-1
votes
2answers
151 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [on hold]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
4
votes
2answers
159 views

How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
2
votes
1answer
65 views

What cardinals do we actually need?

I recently heard of a great variant of set theory called "Pocket Set Theory" in which the only two cardinalities are that of $\mathbb N$ and that of $\mathbb R$ (and the latter is a proper class). ...
2
votes
1answer
96 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
6
votes
1answer
119 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
1
vote
0answers
58 views

Zuckerman's “Sets and Transfinite Numbers”

I am beginning a study in set theory and I found an old book in my school's library by Martin Zuckeman called Sets and Transfinite Numbers which was published in the 1970's. Has anyone used this text ...
2
votes
1answer
113 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
2
votes
0answers
93 views

Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
9
votes
0answers
186 views

Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
4
votes
0answers
49 views

Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition. Definition. Call a set of axioms $\Phi$ in ...
3
votes
1answer
87 views

Has anyone considered axioms to the effect that: “The axiom of constructibility fails very very badly?”

If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that ...
6
votes
1answer
69 views

Conservativity of $\mathrm{ZFC}+\varphi$, where $\varphi$ contradicts CH.

It is well-known that ZFC with the continuum hypothesis is a $Π^2_1$-conservative extension of ZFC. General question. What is known about the conservativity of $\mathrm{ZFC}+\varphi$ over ...
2
votes
0answers
121 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
3
votes
1answer
122 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
3
votes
2answers
149 views

Kunen “Set Theory” 2011 versus 1980 edition - worth buying again?

What are the differences between the original edition (1980) of Kunen's famous book and the new edition (2011)? Is the updated version worth buying? (I hope this kind of question is allowed here. I ...
1
vote
1answer
58 views

Is there a way to phrase “there does not exist a universal set” in structural language?

Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy ...
6
votes
1answer
387 views

Why is ZFC preferred over other set theories?

I was curious why ZFC is preferred over other set theories. Are there specific reasons why? Or is this more of historical reasons?
5
votes
1answer
137 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
5
votes
2answers
80 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
5
votes
2answers
470 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
1
vote
0answers
119 views

Zero sharp and Kunen inconsistence.

There is some (deep/shallow/interesting/trivial) relation between: (1) $\exists 0^\sharp$ iff $\exists j:L\longrightarrow L$ nontrivial elementary embedding. (2) The Kunen inconsistency $\not\exists ...
4
votes
1answer
48 views

Is $2^\alpha=2^\beta\Rightarrow \alpha=\beta$ a $\sf ZFC$-independence result?

In a lecture recently one of my lecturers was proving something to do size of basis or something (I can't remember exactly) and somewhere near the end of the proof we had the following: ...
16
votes
5answers
340 views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
4
votes
1answer
123 views

What determines when an inner model is “canonical”?

I've read several places that L, the constructible universe, is the least canonical inner model. Grigor Sargsyan explains in his slides that L is canonical due to $\mathbb{R}^L$ being $\Sigma_2^1$ and ...
4
votes
2answers
104 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
2
votes
0answers
67 views

Relative consistency proofs using proper class models

Are there any easy relative consistency proofs in set theory that can be done using proper class models rather than set models? The only easy one I can think of is proving the consistency of ...
4
votes
2answers
133 views

A chess problem in Kanamori's “The Higher Infinite”?

Just after Corollary 21.17 (on p289) of Kanamori's The Higher Infinite, he outlines the direction in which he wants to take his discussion of iterated ultrapowers. However, immediately after he ...
0
votes
1answer
58 views

Goodstein's theorem

Recently, a friend of mine introduced me to Goodstein's theorem, which I found to be very interesting and mind-blowing. The theorem basically says that every Goodstein sequence (the wikipedia article ...
19
votes
1answer
1k views

Why is the continuum hypothesis believed to be false by the majority of modern set theorists?

A quote from Enderton: One might well question whether there is any meaningful sense in which one can say that the continuum hypothesis is either true or false for the "real" sets. Among those ...
19
votes
1answer
450 views

Is there any mathematical meaning in this set-theoretical joke?

Recently I heard a joke: If an object exists, mathematicians call it a set and study it. But if an object does not exist, mathematicians call it a proper class and study it anyway. I wonder, ...
3
votes
1answer
83 views

Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
19
votes
1answer
273 views

What are some good open problems about countable ordinals?

After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...
4
votes
3answers
779 views

Does learning logic and set theory before arithmetic, algebra, and geometry have an advantage?

I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic ...
16
votes
5answers
735 views

Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
8
votes
1answer
171 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
5
votes
4answers
275 views

What pattern does set theory study?

If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between ...
2
votes
2answers
76 views

Proving theorems about ZFC by proving them for an arbitrary model.

To prove that a statement follows from the group axioms, we typically write: Let $G$ denote an arbitrary group... Then... Thus, it s a theorem of the group axioms that... Presumably, this form ...
3
votes
1answer
498 views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
2
votes
1answer
102 views

Successor axiom systems and sequences of axiom systems

Let $A$ denote a system of first-order axioms. Is there a canonical way to form a successor system $A'$ extending the ontology of $A$ to include all definable collections? Edit: Importantly we want ...
2
votes
1answer
184 views

Hydra game and quantum superposition

Goodstein's theorem is not provable in Peano Arithmetic showed by Kirby and Harrington in 1982 [Wolfram Mathworld]. Any reference of a "quantum" hydra game where a head can remain in a state of ...
7
votes
1answer
234 views

Is it possible to avoid redundancy in a foundational work?

Imagine we're developing all of mathematics from scratch. We settle on using a set-theoretic foundation. Early on, we assert that an ordered pair $(x,y)$ can be abbreviated $xy$ whenever there is no ...
4
votes
3answers
2k views

Does the set of all sets that contain themselves contain itself?

We always hear about the paradox of the set of all sets that don't contain themselves and whether it contains itself or not. What about the set of all sets that do contain themselves? Is that an ...
9
votes
4answers
428 views

How to introduce advanced set-theoretical objects to philosophy students?

First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce ...
6
votes
1answer
245 views

Set theorist as a physicist or physicist as a set theorist?

I'm majoring physics, but really interested in mathematics. I liked physics since it was really beautiful to have an analysis on a nature with mathematical tool. However, the more i study, the more ...
9
votes
4answers
212 views

Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
3
votes
1answer
329 views

Importance of Kripke–Platek set theory

What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.
3
votes
1answer
291 views

Visualizing Infinity discerning countable and uncountable

This is rather a philosophical question. Although it uses topological notions, it isn't any precise mathematics, so maybe one cannot take it very seriously. Sometimes I try to picture an infinite set ...
3
votes
1answer
400 views

What does the continuum hypothesis imply?

Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom? I'm sorry if this question was already asked. I'm also sorry if there ...
1
vote
2answers
395 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
21
votes
9answers
2k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...