Tagged Questions
4
votes
0answers
58 views
Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof
What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
5
votes
2answers
125 views
The Axiom of Choice and definability
I've seen a lot of relations between the notion of the existence of a definable set with a given property and the necessity of AC is proving that there is a set with the property. For example:
Under ...
9
votes
1answer
105 views
Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
5
votes
2answers
171 views
Intuition behind the Axiom of Choice
Why is it different to make one choice or many choices than to make infinite choices from a theoretical point of view in which indeed you are not going to do any?
How could that be different from ...
8
votes
1answer
107 views
Failure of Choice only for sets above a certain rank
Let $\alpha$ be an ordinal. How can we show that the following theory is consistent?
$\mathrm{ZF}$ + "there exists a set with rank greater than $\alpha$ that is not well ordered" + "every set of rank ...
11
votes
1answer
156 views
Do you need the Axiom of Choice to assert that every real vector space has a norm?
Math people:
This question is 95% answered (the first answer) at Does every $\mathbb{R},\mathbb{C}$ vector space have a norm? and Vector Spaces and AC . The questions, answers, and links found there ...
12
votes
0answers
101 views
Is Dover publishing Moore's book on the Axiom of Choice? [closed]
Dover is publishing a paperback edition of Gregory H. Moore's Zermelo's Axiom of Choice: Its Origins, Development, and Influence. It's supposed to come out March 20th and is available for pre-order at ...
8
votes
2answers
255 views
The “it's not possible” statement in math and the Axiom of Choice
This question actually consists 3 related pieces of text, which I've gathered under this title about which I would like your opinion (they rather contain the implicit question "is this the right way ...
8
votes
4answers
406 views
Why isn't this a valid argument to the “proof” of the Axiom of Countable Choice?
I am having a little trouble identifying the problem with this argument:
Let $\{A_1, A_2, \ldots, A_n, \ldots\}$ be a sequence of sets.
Let $X:= \{n \in \mathbb{N} : $ there is an element of the set ...
8
votes
1answer
243 views
Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?
Math people:
It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. ...
4
votes
2answers
81 views
How does Fraenkel's urelement proof show choice is independent of ZF?
I understand the actual proof Fraenkel gives but I can't see how it proves choice independent of the full ZF because he works in a very restricted universe. Can anyone show how to connect one to the ...
1
vote
2answers
95 views
Question on independent equivalent set
On Page 28, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed),
Say that a set $\Sigma_1$ of wffs(short for well-formed formulas) is equivalent to a set $\Sigma_2$ of wffs iff for any ...
7
votes
3answers
193 views
Does negation of Axiom of Choice imply symmetry?
It seems that every construction of a model in which the Axiom of Choice fails involves some kind of symmetry. Is there an example of a construction of a model where AC fails but no argument involving ...
4
votes
1answer
99 views
Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?
Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...
2
votes
2answers
164 views
Is the negation of Gödel's completeness theorem consistent with $ZF$ without AC?
The proof of compactness and completeness of $\mathscr{FOL}$ (with Hilbert system) used Zorn's lemma. And Zorn's lemma is equivalent to the Axiom of choice in $ZF$.
So my question is can they be ...
6
votes
1answer
277 views
Intuitionistic Banach-Tarski Paradox
While the Banach-Tarski paradox is a counter-intuitive result which requires the Axiom of Choice, leading some people to argue specifically against Choice, and others to argue for constructive ...
3
votes
1answer
98 views
Does (Infer $\phi$ from $\psi$) imply (Infer $\phi^L$ from $\psi^L$)?
I am studying set theory on my own on Drake's famous book and I'm stuck on the (finitary) prove of the relative consistency of the Axiom of Choice.
Is it true that a if we were able to infer $\xi$ ...
7
votes
4answers
312 views
Number Theory in a Choice-less World
I was reading this article on the axiom of choice (AC) and it mentions that a growing number of people are moving into school of thought that considers AC unacceptable due to its lack of constructive ...
22
votes
2answers
495 views
A few questions about intuitionistic mathematics
I have to write a paper on Intuitionism for my Philosophy of Science class and I'm struggling with a few concepts I have encountered in my self-study.
The (intuitive) characterization of valid ...
12
votes
1answer
172 views
Are sets constructed using only ZF measurable using ZFC?
Suppose $S$ is a subset of $\mathbb{R}$ which can be defined without using the axiom of choice, i.e. which can be proved to exist using only the axioms of ZF. Does it follow that $S$ is measurable?
...
3
votes
1answer
168 views
Proof with axiom of choice implies proof without?
Is there a theorem that guarantees the existence of a proof not using AC given there is a proof using AC, at least under some circumstances? What is its name (if there is one) and its most general ...
11
votes
4answers
415 views
Implication and Interpretation of Banach Tarski
As I understand, the Banach-Tarski paradox says a ball in 3-space
may be decomposed into finitely many pieces and reassembled into two balls each of
the same size as the original. Despite being called ...
4
votes
2answers
254 views
Open Sets of $\mathbb{R}^1$ and axiom of choice
In the proof of 'Every open set in $\mathbb{R}^1$ is a countable union of disjoint open intervals', we need to pick one rational representative from each of the intervals hence establish the ...
3
votes
2answers
321 views
How do I choose an element from a non-empty set?
Suppose I have a non-empty set $A$.
How do I choose an element $x\in A$?
More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is ...
9
votes
3answers
697 views
Axiom of choice, non-measurable sets, countable unions
I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
6
votes
1answer
221 views
Simple functions and axiom of choice
The question I have is more of a curiosity, and that is why I decided to post here instead of Mathoverflow.
Before posing the question, let me set up some background.
Background: Let $\Omega$ be a ...
2
votes
3answers
289 views
Truth Value of Theorems in Axiomatic Set Theory
I encountered set theory these past couple of days in discrete mathematics, and my professor was talking about the axiom of choice and ZFC. He said that depending on which axiom you started from, you ...
15
votes
1answer
592 views
Infinite Set is Disjoint Union of Two Infinite Sets
A finite set is a set such that there exists a bijection from it to some finite ordinal. An infinite set is a set that is not finite.
In ZF, can you prove that every infinite set is the union of two ...
5
votes
1answer
351 views
Picking from an Uncountable Set: Axiom of Choice?
Question: Given the real numbers as a set, does it require the (non-finite) Axiom of Choice to pick out an arbitrary single element? What about if we wanted to pick out an integer? What about if we ...
12
votes
3answers
505 views
Where do we need the axiom of choice in Riemannian geometry?
A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the ...
