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30
votes
2answers
5k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
82
votes
15answers
14k views

Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...
18
votes
2answers
1k views

Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...
10
votes
1answer
257 views

Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...
41
votes
2answers
3k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
28
votes
7answers
5k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
9
votes
1answer
461 views

Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...
16
votes
2answers
635 views

If ZFC has a transitive model, does it have one of arbitrary size?

It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote ...
65
votes
9answers
10k views

Solutions to the Continuum Hypothesis

(A related MO question: What is the general opinion on the Generalized Continuum Hypothesis? ) Background The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$. ...
40
votes
13answers
4k views

Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...
25
votes
5answers
4k views

Completion of ZFC

I attended a talk given by W. Hugh Woodin regarding the Ultimate L axiom and I wanted to verify my current understanding of what the search for this axiom means. I find it to be a fascinating topic ...
15
votes
2answers
2k views

Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
14
votes
3answers
2k views

Is there a computable model of ZFC?

Background Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being ...
29
votes
6answers
2k views

Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
26
votes
0answers
809 views

Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ...
14
votes
4answers
3k views

What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...
8
votes
2answers
749 views

Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?

Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...
22
votes
4answers
2k views

What axioms are used to prove Godel's Incompleteness Theorems?

I understand Godel's Incompleteness Theorems to be statements about effectively generated formal systems, which basically makes them theorems about algorithms. This is cool, because despite being ...
16
votes
2answers
988 views

What is a Choice Principle, really?

This question is quite soft, and I apologize in advance if it borderline off-topic. When working in theories between ZF and ZFC the term "choice principle" is heard quite often. For example: $\quad$ ...
13
votes
1answer
367 views

Unpublished works of Woodin on SCH and Radin forcing

There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many ...
11
votes
3answers
901 views

On statements independent of ZFC + V=L

Let $V=L$ denote the axiom of constructibility. Are there any interesting examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? ...
24
votes
3answers
1k views

Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
17
votes
2answers
1k views

A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...
15
votes
1answer
820 views

Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
8
votes
1answer
179 views

Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
7
votes
2answers
2k views

Definition of Function

What is authoritative canonical formal definition of function? For example, According to Wolfram MathWorld, $$isafun_1(f)\;\leftrightarrow\; \forall a\in f\;(\exists x\exists y \;\langle x,y\rangle ...
10
votes
2answers
500 views

What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals? Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...
4
votes
2answers
386 views

Mutually generics

Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...
111
votes
26answers
12k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
52
votes
3answers
5k views

Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
46
votes
12answers
8k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
32
votes
8answers
4k views

Succinctly naming big numbers: ZFC versus Busy-Beaver

Years ago, I wrote an essay called Who Can Name the Bigger Number?, which posed the following challenge: You have fifteen seconds. Using standard math notation, English words, or both, name a single ...
22
votes
15answers
4k views

What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
29
votes
13answers
3k views

Cardinalities larger than the continuum in areas besides set theory

It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably ...
26
votes
2answers
2k views

What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...
22
votes
3answers
3k views

Is “compact implies sequentially compact” consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
18
votes
3answers
2k views

Clearing misconceptions: Defining “is a model of ZFC” in ZFC

There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with ...
9
votes
10answers
3k views

Set theory and alternative foundations

Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set ...
23
votes
4answers
6k views

Non Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...
15
votes
6answers
1k views

How can category theory help my research in set theory?

How can category theory help my research in set theory? I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at ...
14
votes
6answers
2k views

Does finite math need the Axiom of Infinity?

A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects. For example, "The set of primes is infinite" <-> "There is no largest prime". ...
19
votes
6answers
2k views

What is the general opinion on the Generalized Continuum Hypothesis?

I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this. From what I've seen, model theorists and logicians ...
17
votes
5answers
3k views

Set theory and Model Theory

This question probably doesn't make any sense, but I don't see why, so I ask it here hoping someone will illuminate the matter: There is this whole area of study in Set Theory about the consistency, ...
13
votes
8answers
3k views

Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to ...
29
votes
1answer
2k views

Does $2^X=2^Y\Rightarrow |X|=|Y|$ imply the axiom of choice?

The Generalized Continuum Hypothesis can be stated as $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. We know that GCH implies AC (Jech, The Axiom of Choice, Theorem 9.1 p.133). In fact, a relatively weak ...
25
votes
2answers
868 views

Does Fermat's last theorem hold in the ordinals?

My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ where $n\gt 2$, and where we use the natural ordinal ...
22
votes
7answers
2k views

Set theories without “junk” theorems?

Clearly I first need to formally define what I mean by "junk" theorem. In the usual construction of natural numbers in set theory, a side-effect of that construction is that we get such theorems as ...
22
votes
1answer
1k views

Axiom of Symmetry, aka Freiling's argument against CH

Hello! I would like to open a discussion about the Axiom of Symmetry of Freiling, since I didn't find in MO a dedicated question. I'll first try to summarize it, and the ask a couple of questions. ...
30
votes
1answer
2k views

Dual Schroeder-Bernstein theorem

This question was motivated by the comments to Dual of Zorn's Lemma? Let's denote by the Dual Schroeder-Bernstein theorem (DSB) the statement For any sets $A$ and $B$, if there are ...
16
votes
6answers
2k views

Reasons to believe Vopenka's principle/huge cardinals are consistent

There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...