To whom is may concern,

We believed that as the value of the prime number increases, the frequency of prime number occurences decreases. We know that prime numbers grow "rarer" or appear at farther furthered intervals as their value increases. We also know that there exists an infinite amount of prime numbers. If the frequency of primes decreases as the value of the prime increases approaching infinity, then mustn't it be that the rate of prime occurrence must infinitesimally approach zero while/(for) as long as this inversely proportionate relationship persists? Therefore, unless for no apparent reason whatsoever except for perhaps this very conjecture that the frequency of primes randomly becomes either a) unexpectedly unpredictable due to a sudden increased rate of occurence as prime value still continues to increase after some point and then there-ons or b) unexpectedly predictable by way of equidistant prime occurences at regular intervals after some point and then there-ons, or has ever satisfied either of these as qualifying conditions, then it is certain that there must exist a greatest/largest "terminal" or final prime number after which another prime number does not and will not ever exist to occur.

the conjecture: if the limit of or on the rate of the generation of new primes is approaching or approaches zero as the limit of or on the value of new primes is approaching or approaches infinity, then there must exist a terminal prime and the set of all primes must therefore be a finite set.

consideration: if the limit of or on the rate of generation of new primes occuring is approaching or approaches zero as the limit of or on the value of those primes is approaching or approaches infinity, then there must exist an interval of infinite duration during which time no new prime number will occur.

conjectured corollary: consider allowing the limit of or on the rate of generation of new primes approach negative infinity as the limit of or on the value of those primes is approaching or approaches infinity. What might be thereof or therefrom be conjectured?

I also posit that |0| = ∅ = {} = -|∞|. Or, if I may be so bold to modify the notation in a creative way, 0 = }∅{= (∅ - [{ + }]) = -|∞| or ("zero is equal to an or the unbound empty set which is equal to the empty set minus parametered set limitation(s) which is equal to a(n) or the negative absolute infinity").

Thank you for your consideration.

u/PicriteOrNot conjecture: "the primes never become arbitrarily sparse"

1)Show that there is a bijection between the set 2^(N), the set of functions of N in {0, 1},And the set (2^(N))^N, the set of functions from N in 2^(N) 2) Let A and B be sets. Show that if there is an injection from A into B, then there is an injection of A^(N) in B^(N) 3) Use the Cantor--Schr¨oder-Bernstein theorem to show that there is an injection of N^(N) AND 2^(N) 4) Use the Cantor--Schr¨oder-Bernstein theorem to show that there is a BIJECTION between N^(N) AND 2^(N)

Hello everyone, currently I am trying to write a restriction based on a set that I have, and for some reason I have been struggling a lot.

So I have a set L= Ls U Lio U Lpp U Lpd

And the restriction I want to write is that a location lets say Y can be either on Lio or Lpp . (not both at the same time) How I can write that?

Thank you in advance!

Discrete Math

My answers state that x is in the set of integers rather than the set of real numbers. Thus, I think the questions are framed incorrectly because they ask for all x in the real number set that fulfill a certain contingency that only involves integers. I know that the integers are a subset of the reals so I guess it is valid. Honestly, the question is quite straightforward but I wanted to see if I was correct in correcting the inconsistencies of the question.

https://preview.redd.it/248lp4jjied91.jpg?width=3024&format=pjpg&auto=webp&s=a278128f19ff8b3da04b3d8fee16eaed08c634cd

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https://preview.redd.it/7jp0uc4nied91.jpg?width=2944&format=pjpg&auto=webp&s=88af0fe5a06d0acff84bdf185e0a36bb399fa1a6

a) I say that 2 is in the set S for S is the set of all integers greater than 1

b) I say that 2 is not in the set S for S is the set of all integers x that are the square of an integer y

Thank you!

My sister asked me and I dont even know where to start. Makes me doubt I ever even understood it myself

Russell’s Paradox usually defines a set B={x| x∉x}. I thought of an alternative formulation that proves something potentially interesting. The proof is below:

1. ∃x∀y (y∈x<—>y∉y)

2. ∀y (y∈a<—>y∉y)

3. a∈a<—>a∉a

4. a∈a & a∉a

5. ⊥

6. ⊥
7. ∀x∃y(y∈x<—>y∈y)

Since most standard set theories don’t allow sets to contain themselves, this seems to imply that for every set A there is a set B that belongs to neither A nor B.

Besides coming up with an axiom system that nontrivially resolves cardinal arithmetic, my other main hope in studying set theory is inventing new large-cardinal forms. So far, I'm not sure I've come up with any stable such concepts, though; maybe some of my axioms establish some large cardinals of a novel kind, yet they don't seem to be anywhere near as large as I would like (i.e., way beyond measurable).

Until now, maybe, but IDK. Anyway, here are the options:

Background assumptions. Normally, the reflection principle is stated and applied in a somewhat informal (or even 'cavalier') way; I'm sure this or something to comparable effect has been done before, but here, that principle is formalized in terms of a unary operator on generic possible set-descriptive formulae. So I write Я(S), where S is a set-theoretic descriptive formula of the relevant kind. Now, I don't know that upwards reflection in normal set theory is supposed to be like an inverse of V-reflection downwards, but here, it is treated as such. So to indicate how this plays out, let's use some "base cases":

1. Я(V = V) = V (trivial case; since it is impossible in normal set theory for any set to be a set of all sets, this property of V cannot be reflected downwards into a set; informal/incomplete proof follows from the classical definition of V in terms of {x | x = x}.) Alt.: V is a universal set, and the only one, "by definition."
2. Я((cf(V) = V) & (℘(x: x < V) < V)) = an inaccessible set. 0 is trivially inaccessible, but the zeroth aleph is nontrivially inaccessible. (1 is weakly inaccessible, incidentally.) It is unclear to me whether 0 or the zeroth aleph is already sufficient to satisfy the given reflection operation; it seems, 'on the contrary,' as if the reflection operation in question, as stated, has no internal filter so as to reflect forth anything less than class-many inaccessible cardinals. Usually, V's uncountability is thrown into the mix to guarantee reflecting forth an inaccessible cardinal greater than the zeroth aleph, anyway.
3. Я(there is a set with no elements): again trivial; the empty set does not upwards-reflect other empty sets. If 0 is interpreted as a limit number ("the empty limit"), perhaps this upwardly reflects to other limits, though.

The two characterization conjectures I have involve nontrivial elementary embeddings, which are a unified method of characterizing large cardinals (or in Paul Corazza's case, even the zeroth aleph, or rather omega). Rather than the common j: M → N formulation of e.e.'s, we will proceed by writing 𝓔(M, N), read as, "(Attempt to) perform a nontrivial e.e. from M into N."

So, my first conjecture might be interchangeable with the talk of Kunen cardinals offered by Asaf Karagila here. I'm not sure. At any rate, let a Kunen cardinal (not necessarily Karagila's representation) be given first through:

Я(𝓔(V, V) = {x | x is inconsistent with choice/foundation/replacement}) = {y | 𝓔(y, y) = {z | z is as such inconsistent}}

In other words, supposing that V has the property such that trying to nontrivially e.e. V into itself yields a cardinal inconsistent with choice/foundation/replacement, we can downwards-reflect this into the existence of a set such that trying to nontrivially e.e. that set into itself would break the Kunen wall, too. Again, this seems to be what Karagila is saying by emphasizing the specific-rank expression of the Kunen wall vs. the universal 𝓔(V, V) expression. At any rate, this conjecture of mine seems either already manifest in the normal mainstream set-theory context, or close to manifest, so I'm not as excited about it as I was before. (Though note that I'm uncertain how the difference between modular class embeddings and set embeddings can play out, here.)

My next definition is more interesting and promising, I hope. Suppose that j: M → N is an n.e.e. with crit(j) = x. Then note that V > x, for all possible x. V is also greater than any successor, powerset, or local replacement set of/over x. So:

Я(V > {x, succ(x), ℘(x), AOR(succ(x), ℘(x)), ...} = X

... such that: 𝓔(M, N) < X, ∀(M, N) [even, then, for M, N = V], etc.

If this characterization is intelligible, does it comprise a counterexample to the thesis that all large cardinals (or rather all Hamkins/Corazza-style "seeds," down to the zeroth aleph) are directly embedding-theoretic in essence? Do we then speak of "the embeddable universe," "the axiom of embeddability," etc.?

I was going over this MathOverflow question, and I don't think I have enough knowledge of the topic to ask my own question there, but anyway, that question is about the implications of ∃j{j: HOD → HOD}, whether they are closer to the implications of j: L → L or to those of j: V → V.

However, other 'possibilities' are crossovers like ∃j{j: V → L} (see here) or, IIRC (I don't remember a citation off the top of my head), j: V → HOD. So my question is simple, and I might find an answer if I delve into search results long enough or whatever, but for now, here goes: what about j: HOD → L?

From the POV of my research interests, my follow-up inquiry would focus on the 'abstract justification' of this embedding axiom, if there is one. For example, just because ∃j{j: V → L} is 'possible,' still, if I understand my reading correctly, the truth of this axiom would wipe out zero sharp, and the 'intuitive consensus' is that zero sharp exists (there is no such strong consensus in general, though among set theorists who themselves might be styled as implicitly advocating for that sort of consensus, this opinion might well be current, I think), so that gives us reason/'abstract evidence' for the untruth of that specific embedding axiom. (So note that the whole issue appears in different apparel in multiversal set theory.) So using a form of 'extrinsic justification,' ∃j{j: HOD → L} could also be judged. On the flip side, in higher-level terms, asserting such an embedding could be glossed as expressing the epistemic commitments of model theory (that is, we are characterizing something very strongly in particular examples of model-theoretic knowledge, viz. our knowledge of embeddings, of HOD, and of L): the epistemic justification that goes into and comes from model theory, regarding large cardinals, is mediated/transferred to the large cardinal axioms at play when and perhaps only when they can be mapped into the 'space' of large cardinal types available modulo j: HOD → L?

Or, at least, this availability correlate is one case among others of satisfactory epistemic justification for various large cardinal axioms (showing that they can exist vs. the relevant embedding contributes significantly to their abstract degree or measure of justification, but merely not showing that they do not exist otherwise does not contribute negatively to this degree). Again, this is all assuming/hoping/wishing that this embedding doesn't zap the sharps. {Because we also have that question to ask: which sharps exist? Why not all of them? But then I expect the power of HOD would not remain as impressively displayed unto us in our cardinal ascent, would it?} Or even if it does, there is some other surprising but appreciable compensation for the 'hypothetical consensus model' of V in terms of compelling solutions to other problems (even, perhaps, by means of first formulating new intriguing problems entirely, and then maintaining the resources for solving those in turn, up to the local model-theoretic limit on provability).

Understanding of course the reals as subsets of the naturals. Why are there no new subsets of naturals in levels higher than L_w1? I'm looking of course for a proof of this fact.

I've searched for an answer, but I haven't found one yet, probably because it's too elementary and short for anyone working with the constructible hierarchy to bother with it, which is itself a slightly advanced topic. But for the life of me I can't see what I'm missing in this elementary issue.

Thanks for reading

I have just completed intro to set theory and I have wondered how many sizes of infinities are there. I know the size of the set of sizes of infinities is greater or equal to א 0 because if we have an infinite group with a size of |A| we can define a group of a size of 2^|A| and we also know about א 0 but do we know if its size א0 or if its size greater than א0. So coming back to the original question do we know how many sizes of infinities are there?

Sorry for bad English and the bad use of math terms English is not my first language

Hi! I'm doing my Bachelor's Thesis on Gödel's constructible universe L.

I'm interested in the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible universe, Marek and Srebrny, 1973. Of course, the issue is deeply linked to the fine structure of L, as studied in The fine structure of the constructible hierarchy, Jensen, 1971.

I'm searching for recent literature on these issues. Especially, since the gaps in L itself seem to have been thoroughly studied long ago, my advisor has recommended me to look for similar results for generalizations or extensions of L, such as L[R], or inner models of large cardinals, such as L[mu]. I haven't found anything yet, and I'd be very grateful for any information or advice you might have.

The standard text, Constructibility by Devlin, doesn't seem to touch upon gaps of this kind. Gap-n morasses are studied, but from what I understood they're an unrelated concept. I'm also aware of modern developments of fine structure theory, such as hyperfine structure, but these seem again to touch only upon Gap-n morasses.

I've also posted this to the Logic subreddit.

Thanks in advance :)

I'm reading A Book of Set Theory by Charles C. Pinter which uses a slightly modified version of NBG (but I think this also applies to normal NBG). Here's the argument:

P1. Everything is either a set or a proper class (one is the negation of the other, so this follows from LEM)

P2. Assume that there is a proper class, A, so ∀Y. ¬A∈Y

P3. Since it is true of all classes it is true of some arbitrary class Y, so ¬A∈Y, which is equivalent to saying A∈Y'

P4. However, it is also true of Y', so ¬A∈Y', which is the same as A∈Y

P5. A∈Y and ¬A∈Y

C1. Therefore, there no proper classes and everything is a set

C2. The reason given for why the russell class (i.e. {x|¬x∈x}) didn't lead to a contradiction was because it wasn't a set. By this proof, it is a set so the contradictions remain

I'm at the final section of chapter 1 in the book and everything up to that is basically all I know about set theory. This seems really obvious because even a newbie like me noticed it so someone smart must have responses for why this is wrong that I don't know about. I have no idea if this has a name so I have no idea what to google (even what I tried to google didn't give me anything). Any help?

I was reminded of a result I first saw in Mathematical Problems and Proofs, that shows that the generating function for the Fibonacci Sequence implies that the sum over all Fibonacci numbers is -1.

There's a very good proof here in the first response to the question:

The generating function for the Fibonacci numbers

Simply set 𝑧 = 1, and you have the sum in question, which is plainly -1.

Of course, the sum diverges, which initially lead me to view this result as a mere curiosity, though it just dawned on me, that there's an additional problem, that I think is tantamount to a paradox:

Let 𝑆 be the set produced by the union of disjoint sets of sizes 𝐹1,𝐹2,…, where 𝐹𝑖 is the i-th Fibonacci number. It must be the case that 𝑆 has a cardinality of ℵ_0. More troubling, addition between positive integers can be put into a one-to-one correspondence between unions over disjoint sets, and thus, we have an apparent paradox.

Note this has nothing to do with convergence -

In fact, the problem is that the sum is a finite number, whereas a perfectly corresponding union over sets produces the correct answer, of ℵ_0.

Does anyone know of a resolution to this?

One initial observation, addition is not commutative with an infinite number of terms, though I'm not certain at all that this is what's driving the apparent paradox, but it does show that the rules of algebra must change with an infinite number of terms.

In this essay, Evans and Hamkins explain how to characterize all countable ordinals as game values in 3-dimensional infinite chess. At first, I just wondered what that would mean modulo CH (since the set of all countable ordinals is ℵ₁) but a much more interesting question occurred to me: since all proof-theoretic ordinals are countable, then is there some way to mark them out using this chess-theoretic characterization?

I imagine there'd be two ways:

(A) use something like "recursive chess theory" to directly mark them out.

(B) characterize a large countable original (in excess of the proof-theoretic threshold) in chess-theoretic terms, then use an ordinal collapsing function to mark out the relevant proof-theoretic ordinals.

It's taken me two years of independent study to get even moderately good at set theory in general, though; I am not versed in the art of ordinal analysis. So, I'm not in a position to pursue these ideas in too much detail, by myself.

BONUS QUESTION: from what I've heard, we're ages from figuring out ZFC's proof-theoretic ordinal. Might any of this help out with that?

Let:

U={c, d, g, l, p, q, t}

X={c,g, p, t,} Y={c, d, g}

Z= {d, g, l, p, q}

Find the following set: (Z U X') 'n y

(In this Find section of the equation these are not the letter U but union and intersection)

[Foreword: references to the proper class J are to be understood in terms of my MathOverflow question about this proper class.]

You can vary your set theory by using different axioms, or you can use a different logical background altogether. So we have things like CZF or Zach Weber’s paraconsistent set theory. These alternatives are not without intriguing consequences: for example, it has been said that second-order logic grounds a system in which the question, “Is CH completely resolvable?” admits of the answer, “Yes” (although we do not yet know what the specification of this “Yes” would ultimately come down to). Or suppose that you said that the Continuum is “simultaneously” equal to every aleph that it can be forced to equal: this is a paraconsistently admissible hypothesis, and a clever philosophical argument (from the justification-theoretic side of forcing) might be designed on its behalf moreover, wherefore… And then there is the modality-theoretic gloss of universal questions in the Hamkins cosmos. In the spirit of the above variations, I would like to present a set theory in which the logical background is deontic; the hope is that this theory can offer novel and well-justified solutions to an array of problems in the field.

The elementary idea

The easiest way to have some form of DST is to have ought-sentences in the system. Let OB(S) read, “It ought to be that S.” So now we might have OB(x is an element of y), for example. The choice between using the OB-function as a propositional operator or as a predicate is quite important as a logical issue on its own, but for the purposes of DST at present, I will waive the issue and say that we could also have sentences like, “X ought to be an element of Y,” and these capture what DST is meant to be, on an elementary level, just as well as if we only used OB as a propositional operator.

The immediate corollary is: to get deontic ZFC from normal ZFC, take various is-sentences from the latter and translate them into ought-sentences for the former. So now we also have assertions like, “A ought to be the powerset of B,” for example. In the standard expansion of ZFC, we also have nontrivial elementary embeddings in general, so now we also have something like OB(j: M → N).

There’s a saying, “Moral responsibility requires free will,” and one theory of free will requires a specifically given contingency (the “principle of alternative possibilities”), so if we adopted that theory, we would have to say that in DST, there are abstract sets who contingently have their elements. (Either they already have these elements, and there is at least one person somewhere in some world, who has the ability to take these elements out of those sets; or the sets are yet waiting to have their elements added into them, wherefore the fact is to be stated as, “There are sets that ought to have elements added to them.”) This claim can be made to fit onto the Zermelian universe (the “unfinished totality” description of V) or the Hamkins multiverse: here are Brouwer’s “freely creating subjects,” but with far more power than Brouwer expected those beings to ever have. Other than making note of that, though, I will waive the matter of contingently full sets: even if the deontic sets can “change size” at will(!), we will consider them as if they were, in Cantor’s paradise, safeguarded from such changes.

Observation. Model-theoretic reflection justifies some mathematical beliefs. For example, there are theorems in model theory, the consciously processed proofs of which count as justifications for them. Or model theory gives us the resources to well-form our definitions of various large cardinals, and even if definition-sentences might not necessarily seem "justification-apt" (you either accept the stipulation or you don't?), I will claim that to well-form a definition is to at least partly justify it. Alongside the theory of the proper class J, this all means: there is an axiomatic set theory of the justifiable universe; for every axiomatic set theory of a universe, there is a first cardinal that yields a model of that theory; so there is the equivalent of a basic "worldly" cardinal for deontic ZFC. {Later, I will more strongly claim that this model-theoretically identified aleph is inaccessible-from-below, too.} Call this cardinal D. Inverse adduction: assume that deontic ZFC has a proof-theoretic ordinal assigned to it. Assume that this ordinal can, at least in principle, be identified via an ordinal collapsing function. Now these functions take large countable ordinals for their inputs to do their work, so now we have a large countable ordinal that has something to do with deontic ZFC as such in place. But these such large countable ordinals can be represented as special counterparts of genuine large cardinals. Presto: there is a deontic large cardinal: again, call this D.

Special properties assigned to the deontic cardinals

As part of the definitional quest, I have also considered stipulating that the deontic cardinals, whatever else they primarily are, are such as to have various secondary properties that set them apart, in relatively important ways, from other large cardinals. There is a simple argument for the first of these stipulations, so I will go over that one first. Now, shift to the logical metaethics of DST and hold that, if ethical information is to be sufficiently action-guiding, it must be accessible; overly complicated proofs or unattainably many samplings do not figure in the rational answers to ethical questions. (E.g., if you have to write a 200-page deduction to conclusively answer an individually practical ethical question, then even if you have the right answer, you have it in the wrong way.) Accordingly, the arithmetic of the deontic cardinals must be resolvable in a way that the arithmetic of the other transfinite cardinals is not: whereas it is impossible to prove what the powerset of the zeroth aleph is in normal ZFC, in deontic ZFC, by contrast, it should be possible to prove what the powerset of D is. In slogan format: “Modulo DST, the powerset of D ought to be resolvable,” and, “Modulo DST, X [whatever it turns out to be] ought to be the powerset of D.”

As far as I can tell, this stipulation in no wise establishes that D, et. al. are intrinsically greater than any large cardinal heretofore well-defined. Whether they admit of a resolvable arithmetic does not seem to be a property of commensuration with the other alephs. However, the next exotic property assigned to the deontic alephs does affix them quite far up the universal hierarchy.

First, assume a reflection principle such that, for every special property (appropriately categorized) that V has, there is a first set-sized domain that shares this property. The Kunen wall determines such a special property: for this is the fact that trying to perform a nontrivial elementary embedding from V into itself will yield a hyperset structure that conflicts with the axioms of choice and foundation. So there is a first set X that shares the property of the Kunen wall, i.e. is such that trying to perform a nontrivial elementary embedding from X into itself would break that wall. Now, the background theory of justification allows us to admit hypersets into our universe, so we will now assert that the Kunen wall is indeed broken when one nontrivially embeds the elements of D into D itself, that is D is as such keyed to at least one corollary hyperset; but in fact, every deontic cardinal that has the property of the wall is keyed to some hyperset; so the deontic cardinals open the doors in Cantor’s paradise, from the ascending domain of the alephs to the vortex and nexus of the descending justification numbers.

Less poetically: in established set theory, there are “translation methods” that allow you to take a strictly well-founded theory and justification-theoretically key it to a hyperset or loopset system. (At least, I think that's what Hamkins was telling me, in his reply to my MathOverflow post.) But in DST, we don’t just have justification numbers, but justification values more abstractly. The precedent for this concept is Frege’s doctrine of truth values. Anyway, the forms of justification proper (foundationalism, coherentism, and infinitism) therefore color the very semantics of the set theory in play, so the hyperset section of V (the proper class of all infinitist sets), modulo V = J, is semantically distinct from the well-founded and loopset sections, etc. To refer to one section is not to directly refer to the others, as such. Now, on top of all that, the eternal sequence of inference in question, is one given via the general ability to ask questions itself: the proper class of regressive erotetic inferences, from the axioms of the universal hierarchy, constitutes the given universal hyperset itself, here, i.e. our knowledge of the universal erotetic regress justifies our assertion that there is a universal hyperset, but this is the only justification offered for such an assertion so far. So far, this is the only universal hyperset we get to work with in DST. Accordingly, if the deontic cardinals are affiliated in a peculiar way with the possible outputs of the justification function, we can say that the hypersets to which the deontic cardinals are embedding-theoretically keyed are parts of the universal hyperset. It is through the deontic cardinals that the hyperset section of V is fully unified with the ascending hierarchy, not through a neutral translation schematic. But this unification only goes through if we attribute the property of the Kunen wall, to the deontic cardinals.

Undecided description: let L(D) be a constructible universe from D upwards. Is it permissible, in mainstream set theory, to have a universe that is not "always" L-like, yet "sometimes" it is, too? What I would like to say is that, regardless of whether V = L below D, as of D, V = L(D). But I don't know whether that's logically doable. If it is, my intuition is telling me that there's a good chance it would be doable at least in some kind of Hamkins multiverse. {The internal reasoning for L(D): if L-like systems are always the "simplest" out of their relevant lists of alternatives, then by the axiom of deontic simplicity again, we have DST recommend attributing L-likeness to the deontic hierarchy, inasmuch as the simplicity of L makes resolving D-questions more tractable as such. In fact, we can go back to what we said about knowing what the powerset of D ought to be (about how we ought to know this!), and say that the derivation of GCH from the L-theoretic sets is given again for the D-theoretic ones, as fully justified. (By contrast, the normal axiom of L is not fully justified, if it is justified almost at all! Case-in-point: if this axiom were true, there go the measurable cardinals.) The model-theoretic simplicity of L is adapted to the requirement on epistemic simplicity, in deontic logical space.}

Side-note: the inaccessibility question. In the set-theoretic mainstream, axioms of inaccessible cardinality admit of reliable intrinsic and extrinsic justifications. Nevertheless, no such axiom is accepted as strongly as the axioms of ZFC proper are. And of course, nowadays, set theorists are less likely to worry about justifying inaccessibility axioms than they are axioms involving much stronger concepts of the higher infinite.

However, I would like to go over the idea that the inaccessibility question, because of the way it is posed relative to ZFC proper, admits of a rigorously deducible answer, modulo DST. Firstly, then, ZFC actually already encodes two inaccessible cardinals, zero and the zeroth aleph. For reasons of lexical tradition, however, we usually don't say that those numbers are inaccessible, because we're really wondering about uncountable inaccessibles, not countable ones. A third inaccessible, if it exists, must be so large that it yields a model of ZFC, after all. Anyway, though, I think we should differentiate between what ZFC can prove and what it can justify. Since I am using an erotetic logic here (again see the linked MathOverflow post), I will say that there is an erotetic method of justification, such that, if a sound theory evokes a given question, then this fact of evocation somehow justifies some further assertions, in the relevant context. By way of example: the presence of the two inaccessibles, in ZFC space, directly evokes the question, "How many inaccessibles are there altogether?" And the mere existence of this question somehow justifies answers like, "Possibly X-many altogether..." for some X.

Granted, of course, this question cannot be decisively answered in ZFC. Moreover, the usual approach is to assume answers to this question axiomatically: e.g., we assume that there is one uncountable inaccessible, or X-many (for some finite or transfinite cardinal X), or class-many. If we assume the existence of some other type of large cardinal, which incidentally codes for inaccessibility to boot, then we can get sentences like, "For the initial K measurable, there are K-many inaccessibles below that initial cardinal," which is at least a partial special answer to the inaccessibility question. I think but can't prove that it would be easy to show that D is inaccessible; so, "At least three," becomes the answer to our question, here. However, if D is inaccessible, it seems to me "likelier" that there are other inaccessibles below D, besides the original two. What I would like to show is: how many exactly? And then: how many, if any, are inaccessible above D, as well? And I would like to show these things, properly: I don't want, "There are 24, or 72, or aleph-24, or aleph-72, or however many, inaccessibles in total; the amount is to be assumed." I want the answer as a theorem. {Caveat: part of me would like to stipulate, as another secondary characteristic of theirs, that the deontic cardinals do not enter into the inaccessibility relation amongst themselves, i.e. D is inaccessible from below, but from D, every higher cardinal is eventually accessible. For some reason, inaccessibility "runs out" as of D. However, I can't properly motivate this stipulation, for the time being; we might try out the saying, "All alephs above D, ought to be eventually accessible from D," but why "ought" this be so?} {The general saying, "There is an initial segment of V in which all alephs ought to be accessible from the initial interval of that segment onward," might be grounded in abstract consistency, i.e. if this saying does not internally or externally violate the law of noncontradiction, it is admissible; so we would have this realm of deontic accessibility in general, which might be safely transposed over the already-delineated deontic realm. But it would also be possible to have the D-successors and limits comprise an entirely lower realm, as such, so that we have D-inaccessibility up to some other point, D say, so that it is from D onward that inaccessibility vanishes.} {Note that if inaccessibility vanishes as of D or D, it would follow that we could bracket our answer to the inaccessibility question like so: "At least 3, and at most D (or D)."}

So although I can't say for certain that deontic ZFC, as it so far stands, encodes such an answer as a theorem, I will say that deontic ZFC, if necessary, ought to be revised, so that the inaccessibility question becomes a matter of theorems and not axioms as such. But more importantly, for present purposes, that D is inaccessible shows that DST provides an extremely strong justification for asserting that uncountable inaccessible cardinality exists in the first place. And DST, as a crystallization of justification-theoretic logic overall, inherits an extremely strong justification for a kind of worldly cardinal, too. So DST strongly justifies asserting the existence of various large cardinals, and hopefully can be refined so that it asserts a specific amount of this existence (so to speak), too. In other words, the large cardinal axioms present in DST, should be candidates for acceptance on the same level as the standard ZFC axioms are on, I think. {Arguably, the way that sets are given in the context of J altogether, establishes that J-theoretic existence sentences are almost by definition justified; so perhaps it is true that the D-inaccessibility axiom is even more justified than the normal ZFC axioms, no less?} {Note that the plenitude-theoretic method of cumulative ontology, according to which a set of ontological sentences (in set theory) is justifiable enough just in case those sentences are consistent, while not denied here, is not advanced, either; that is, we are proposing a different and specific ontological method (deduction from the J-facts}.}

WRAP-UP

In this analysis, I hope to have accomplished the following:

(A) Legitimately characterized (provided a well-formed definition of) a new type of large cardinal. The primary attempts are by model-theoretic and proof-theoretic explanation.

(B) Offered a very strong justification for incorporating the existence of this cardinality (and its ZFC-worldly and inaccessible shadows) into our standard ontology, i.e. the justification is sufficient for adopting the axiom system in question, with as much practical conviction as we have heretofore adopted ZFC simpliciter. It is supposed that otherwise, no known large cardinal axiom, even such as, "There is at least one uncountable inaccessible," has yet to be justified enough to be a candidate for a definitive consensus extension of ZFC. (I admit, this is a strong claim to make, seemingly ignorant of the traditional intrinsic and extrinsic justifications that inaccessibility axioms have, which are, after all, strongly indicative of the existence of even a proper class of inaccessibles. Worse, this talk of "definitive consensus" might seem politically absurd or by now outdated (what with the multiverse hypermodels in play in mainstream set theory nowadays). Maybe you can just look at the issue in these terms: "If there were such a thing as this 'definitive consensus,' then the D-theoretic axioms would be justified enough to merit inclusion in this consensus." We might say that we can at least form an abstract mathematical image of such 'consensus,' no less. How far this image may be applied to historical reality, would be a next issue.)

(C) Indicated that the type of large cardinal here spelled out, would be initially larger than all types conceived by set theorists in the historical mainstream (e.g. the least deontic cardinal is larger than the least 0-huge (measurable) cardinal, etc., up until we reach the omega-huge range and break the Kunen wall; from what else has been said of this matter, it would follow that we could have the deontic cardinals as expressions of omega-hugeness). This is a daunting and ambitious task to have set for myself; I am the least confident in my success on this score. However, I do think that I have strongly indicated that the deontic cardinals do start out quite far up the universal hierarchy, even so: only they might not be the "greatest" of all, but "on a par with" other candidates for such ranges?

Addendum

After posting the above, I was thinking about juxtaposing DST with infinitary logic. At first, I just thought about having ℒ(D, D), meaning: what would be the properties of such an ℒ? However, I was then reading about weakly, etc. compact cardinals, which are posited in terms of infinitary logic. So it occurred to me: if there were an infinitary deontic/justification logic, could there be weakly, etc. compact cardinals defined relative to that logic? Presto: again we get an image of D. Not that D, here, would be any higher up than normal weakly, etc. compacts would be (I don't know that they would). On the other hand, modulo the issue of intrinsic justification, I think we have very strongly intrinsically justified axioms, here: not because they can be "unfolded from the iterative conception of set" or because they satisfy reflection principles (the forms of "intrinsic justification" I am most familiar with from the standard literature), but because they represent the very concept of mathematical justification itself. Anyway, an ℒ-theoretic definition of D, et. al., seems promising, to me, especially as it seems to open the door for a justification of many other large cardinals besides worldly or inaccessible ones.

Hello, I am taking my first ever statistics class and cannot wrap my around the difference between a connected relation and an asymmetric relation.

To me, since they both imply that if aRb then bRa cannot exist, they mean the same thing.

I asked this same question on Quora and was given this answer:

——————————————————————- A relation < is “connected” if for all distinct pairs, 𝑥<𝑦 x or 𝑦<𝑥. But the “or” here permits both to be true.

It is “asymmetric” when 𝑥<𝑦 implies 𝑦≮𝑥. But “implies” permits 𝑥<𝑦 to be false and 𝑦≮𝑥 to be true.

So, a relation could be connected but not asymmetric (if a pair is related in both directions), or asymmetric but not connected (if a pair is not related at all.)

For example, the relation {(𝑎,𝑏),(𝑏,𝑎)} is connected but not asymmetric.

The empty relation {} (over 𝑎,b) is asymmetric but not connected. ——————————————————————

However I still fail to understand this; how can “or” mean that both relations can hold (in the case of connection?) doesn’t or mean the exact opposite of that?

Thank you in advance if you’ll take the time to help me out!

I'm working my way through "A Book of Set Theory" by Charles C. Pinter. (I'm not very far into it yet.) In the Historical Introduction (Chapter 0), he spends a lot of time discussing logical and semantic paradoxes in naive set theory. Then, in Chapter 1 (Classes and Sets), Section 2 (Building Classes) he lists two axioms:

A1: A=B iff (x ∈ A) ⟹ (x ∈ B) and (x ∈ B) ⟹ (x ∈ A).

A2: Let P(x) designate a statement about x which can be expressed entirely in terms of the symbols ∈, ⋁, ⋀, ¬, ⟹, ∃, ∀, brackets, and variables x, y, z, A, B, ... Then there exists a class C which consists of all the elements x which satisfy P(x).

Immediately after A2, he writes the following. "The reader should note that axiom A2 permits us to form the class of all the elements x which satisfy P(x), not the class of all the classes x which satisfy P(x); as discussed on page 13, this distinction is sufficient to eliminate the logical paradoxes. The semantic paradoxes have been avoided by admitting in axiom A2 only those statements P(x) which can be written entirely in terms of the symbols ∈, ⋁, ⋀, ¬, ⟹, ∃, ∀, brackets, and variables."

Here's my question: Does this mean that the equal sign (among other common symbols, like <, >, etc.) is not allowed in P(x)? I should note that later on (on the same page as the quote above), he defines "the universal class U" as "the class of all elements. The existence of the universal class is a consequence of the axiom of class construction, for if we take P(x) to be the statement x = x [the bold is my modification], then A2 guarantees the existence of a class which consists of all the elements which satisfy x = x..." I'm not sure I understand what Axiom 2 is saying. To me it says symbols like "=" are not allowed in a predicate/statement. But then he puts one in there anyway.

Could not find it out so im really curious. Thanks!

I'm going to reframe this in a way that is hopefully different.

If we do not allow for superfluous 0s in the decimal numbers (0.1 is the same number as 0.10). And pair the whole numbers up with a corresponding decimal number by flipping the interget digits over the decimal (1 -> .1, 10 -> .01, 19 -> .91, etc...) then every integer pairs up with every decimal and every decimal pairs up with every integer. Which would prove there are the same number of numbers between 0 and 1 as there are integers from 1 (inclusive) to infinity.

I must be wrong because it seems like a simple exercise that someone else would have thought of but I cannot think of a number on either side that cannot be uniquely represented by the other.

In logic systems for example first order logic, is it correct that they assume pre-existence of structures, and relations and functions on the domains of structures?

For example, the satisfaction relation between structures and sentences in FOL is defined, based on the usual "satisfaction" relation between relations on domain sets and the elements of domain sets. So the usual "satisfaction" relation between relations on domain sets and the elements of domain sets must already be defined before introduction of FOL.

Do definitions of relations and functions on the domain sets of structures, and the usual "satisfaction" relation on the domain sets, require or lead to a certain kind of "set theory" that exists before and without FOL? What is the name of that kind of "set theory"?

ZFC theory and various kinds of set theory are based on FOL and assume that we have already had FOL. So they don't seem to be the kind of "set theory"?

How do logic systems, structures/models, and set theories resolve this "chicken-and-egg paradox": does a set theory come first for defining relations and functions on domains of structures, or does a logic system (FOL) come first for defining a set theory?

What does "naive set theory" mean? Is it a rigorous concept, and an actual theory in a logic system? Is it the kind of "set theory" which is used in domains of structures and exists before and without a logic system?

Thanks.

In ZFC, what is the definition of a "proper class"?

In Jech's Set Theory, he defined a class as {x: phi(x)}, and considered every set S as class {x: x in S}. Then he wrote

That the set S is uniquely determined by its elements follows from the Axiom of Extensionality.

A class that is not a set is a proper class.

Does the above consideration of every set as a class define a mapping from the sets to the classes?

Is the mapping not surjective? That is, is there a class not mapped from any set by the mapping?

Is a class which is not a set a proper class? I think so, but is confused by https://math.stackexchange.com/a/2869598/ which seems to say no:

if a formula defines a class, as any formula does,

if it is not provable that it defines a set, does it necessarily define a proper class? The answer to that is negative.

Thanks.

I wrote this some time ago, and didn't realize there's a subreddit for set theory, and given that the ideas are plainly not traditional, any insights would be appreciated, as I don't know the literature terribly well, and instead approached the topic wearing the hat of an information theorist.

The basic result is, the logarithm of Aleph_0 is an unusual number, that does not correspond to the cardinality of any set, but can be rigorously described as a quantity of information.

https://www.researchgate.net/publication/349913208_On_the_Logarithm_of_Aleph_0

interested in how ZFC spread (at school or academia) in the 19th and 20th centuries, and why it spread so quickly. how did a theory that was essentially crazy nonsense turn into a kind of prima donna once it was put in a context of lacking a formal axiomatic system consistent with inference rules, thereby demanding the creation and fulfillment of the leading role of ZFC in its little hysterical drama. My impression is that the history of ZFC is a kind of Shakespearean tempest in a teapot. It seems a kind of metaphysical obsession with applying “truth” and logic in an if-you’re-a-hammer-everything-is-a-nail way got the better of calculating with expressions denoting sets. My point is that it looks quite silly from a historical point of view to suggest that perhaps the cure for Cantor’s mental illness was to imagine the reality of his imagination by applying the inference rules flying the banner of truth in a kind of military conquest of what was initially crazy nonsense. Why was “curing” Cantorian set theory of its supposed untruth or paradoxical | contradictory aspects considered such a cause for hysterical activity? How to approach this, what research has been done so far, and in particular if set theory followed a viral model of spreading throughout European schools?

Hi, I’m looking to start studying set theory in order to increase my understanding about mathematics. Can someone help me with some basic materials (hopefully PDFs)?

I know all the concepts of set theory. but i want to learn how to code them or how to use algebra to compute set operations. do you guys know any good references? like I know what A union B means and what it does but I dont know how to apply it on data or matrix:(