For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.
12
votes
1answer
519 views
Proofs given in undergrad degree that need Continuum hypothesis?
Or alternative you need to assume CH is false.
I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
2
votes
2answers
270 views
what is the relationship between ZFC and first-order logic?
In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the ...
33
votes
4answers
4k views
Can you explain the “Axiom of choice” in simple terms?
As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get.
I went to Wikipedia to see what the Axiom of ...
20
votes
6answers
1k views
What are natural numbers?
What are the natural numbers?
Is it a valid question at all?
My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
6
votes
9answers
526 views
Motivating implications of the axiom of choice?
What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
3
votes
2answers
265 views
Axiom schema and the definition of natural numbers
An axiom schema is used to generate the axioms, which inductively define the natrual numbers using the empty set and the successor function $S$.
I don't understand why you have to define this set as ...
20
votes
5answers
713 views
How is a system of axioms different from a system of beliefs?
Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
(PD: I'm not religious)
11
votes
6answers
496 views
When does the set enter set theory?
I wonder about the foundations of set theory and my question can be stated in some related forms:
If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not ...
10
votes
4answers
388 views
Axiomatic approach to polynomials?
I only know the "constructive" definition of $\mathbb K [x]$, via the space of finite sequences in $\mathbb K$. It essentially tells a polynomial is its coefficients.
Is there a way to define ...
8
votes
2answers
314 views
Where is axiom of regularity actually used?
Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom?
This question was to some extent provoked by Dan ...
40
votes
5answers
2k views
In what sense are math axioms true?
Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers.
The kid asks: why?
Well, it's an axiom. It's called commutativity (which is not even true for most groups).
How do I ...
9
votes
5answers
439 views
What are the postulates that can be used to derive geometry?
What are the various sets of postulates that can used to derive Euclidean geometry?
It might be nice to have several different approaches together for comparison purposes and for ready reference.
It ...
16
votes
5answers
394 views
Why hasn't GCH become a standard axiom of ZFC?
I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that:
The cardinal numbers ...
5
votes
3answers
174 views
The existence of the empty set is an axiom of ZFC or not?
I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...
3
votes
1answer
110 views
System with infinite number of axioms
Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic ...
3
votes
3answers
148 views
An elementary question regarding the uniqueness of a set, viewed with different cardinality
Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with?
If so, what does it mean to speak of such a set if it is not really one single ...
1
vote
1answer
170 views
Do the proofes in set theory rely on the semantics of the formulas used in the axioms?
Motivation: The Axiom of separation
$$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$
is used to ...
4
votes
2answers
182 views
How does (ZFC-Infinity+“There is no infinite set”) compare with PA?
How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
4
votes
3answers
241 views
Why is the postulate $1$ not equal to $0$ not superfluous? [duplicate]
Possible Duplicate:
Explanation for why $1\neq 0$ is explicitly mentioned in Chapter 1 of Spivak's Calculus for properties of numbers.
I am self-studying the wonderful book, Elementary ...
3
votes
2answers
207 views
What is the difference between an axiom and a postulate?
I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
3
votes
1answer
220 views
Can it be shown that ZFC has statements which cannot be proven to be independent, but are?
I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would ...
3
votes
2answers
466 views
A first order sentence such that the finite Spectrum of that sentence is the prime numbers
The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ ...
2
votes
2answers
109 views
Axioms for sets of numbers
What is the most common axiomatic system used by modern mathematicans for the properties of the integers, rationals, reals, and complex numbers?
Or does one commonly use a single axiomatic system ...
1
vote
0answers
32 views
Using definitions instead of axioms.
Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms.
Given all this, a rigorous work of mathematics will typically begin with a signature - ...
1
vote
1answer
60 views
What makes Tarski Grothendieck set theory non-empty?
I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
1
vote
1answer
119 views
What should I be able to do with this chapter on Axiomatic Set Theory in order to check if I've learned it decently? [closed]
I've just read a chapter on axiomatic set theory, from Comprehensive Mathematics for Computer Scientists 1. It comes with basic notation on sets and some axioms:
Axiom 1 (Axiom of Empty Set)
Axiom 2 ...
0
votes
6answers
463 views
Is it possible to have a field without an additive identity?
If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization?
By definition it is not possible, but are there ...
