For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.
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 ...
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 ...
24
votes
7answers
1k views
Are there infinite sets of axioms?
I'm reading Behnke's Fundamentals of mathematics:
If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.
I got curious on this: Are there infinite ...
23
votes
7answers
1k views
Does mathematics require axioms?
I just read this whole article:
http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
which is also discussed over here:
Infinite sets don't exist!?
However, the paragraph which I found most ...
21
votes
5answers
741 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)
21
votes
6answers
583 views
Axiomatic characterization of the rational numbers
We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this).
...
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 ...
16
votes
5answers
423 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 ...
13
votes
0answers
163 views
Is there an axiomatic approach to ordinal arithmetic?
I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers?
If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
12
votes
1answer
531 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 ...
12
votes
2answers
98 views
How far is it true that statements dependent on Axiom of Choice are not constructive.
Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
11
votes
6answers
513 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
392 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 ...
9
votes
5answers
449 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 ...
8
votes
5answers
683 views
Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?
Context: Rings.
Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?
Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
8
votes
5answers
418 views
What is the modern axiomatization of (Euclidean) plane geometry?
I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
8
votes
2answers
335 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 ...
8
votes
2answers
181 views
Axiomatic system and Hilbert's 2nd problem
Hilbert's second problem asks if the axioms of arithmetic are consistent. Has this problem been resolved? Shouldn't an axiomatic system ideally be consistent and complete(given that we have the ...
8
votes
2answers
246 views
In axiomatization of propositional logic, why can uniform substitution be applied only to axioms?
I'm reading an introductory book about mathematical logic for Computation (just for reference, the book is "Lógica para Computação", by Corrêa da Silva, Finger & Melo), and would like to ask a ...
8
votes
1answer
108 views
What is the smallest fragment of ZFC that has the same consistency strength as ZFC?
The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it.
Thanks to Godel and Cohen, we know that ...
7
votes
5answers
305 views
A confusion about Axiom of Choice and existence of maximal ideals.
The proof of the theorem of statement that every ring has a maximal ideal uses zorn's lemma or the axiom of choice.Now, the defintion of ring as well as the definition of maximal ideal don't depend on ...
7
votes
3answers
315 views
What constitutes an axiom - Spivak Calculus ch. 1
In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a ...
7
votes
2answers
351 views
Relationship between Category theory and Axiomatic set theory
I've recently started learning Category theory- and I have a pondering- wondering if anyone can help.
Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
6
votes
9answers
542 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 ...
6
votes
5answers
258 views
How to demystify the axioms of propositional logic?
How might I go about getting some intuition on the typical axiom schemes given for propositional logic? They seem rather mysterious at first glance.
For example, these are taken from: ...
6
votes
2answers
283 views
Is this formula really the nine axioms?
I was reading a note from guardian.uk called What lurks beneath a scientist's lab coat?, a little gallery of geeky-tattoos.
However, number 11 in the series has the following image and caption text:
...
6
votes
2answers
85 views
Is there a way to axiomatize the category of sets and relations?
The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?
6
votes
1answer
975 views
Euclid / Hilbert: “Two lines parallel to a third line are parallel to each other.”
Background
Many geometry books used to teach high-schoolers these days try to transfer Hilbert's reworking of Euclid's axioms into a (somewhat) palatable form for students. They don't usually seem to ...
5
votes
3answers
580 views
Has anyone ever proposed additional axioms?
According to Wikipedia, Godel's incompleteness theorem states:
No consistent system of axioms whose
theorems can be listed by an
"effective procedure" (essentially, a
computer program) is ...
5
votes
2answers
251 views
What interpretations exists for the singleton of the singleton of the empty set $\{\{\varnothing\}\}$ and its singleton $\{\{\{\varnothing\}\}\}$?
What I know
I know from Peano's axioms that the empty set is equivalent to the natural number $0$ and that the singleton of the empty set is equivalent to the natural number $1$. ( ...
5
votes
3answers
271 views
Are “axioms” in topology theory really axioms?
If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true.
I have seen that in topology theory, various axioms of countability and separation ...
5
votes
2answers
200 views
“Completeness modulo Godel sentences”?
So this has been bugging me for roughly four years. When I was an undergraduate, I attended a colloquium in which the speaker was a 'cheerleader' for AD (the axiom of determinacy- an alternative to ...
5
votes
3answers
197 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 ...
5
votes
3answers
422 views
How do the separation axioms follow from the replacement axioms?
It has come to my attention that the pairing axiom and the separation axiom schema are rarely listed since they follows from the replacement axioms. I see how this works for the pairing axiom, since ...
5
votes
3answers
137 views
The axiom of infinity for Zermelo–Fraenkel set theory
The axiom of infinity for Zermelo–Fraenkel set theory is stated as follows in the wikipedia page:
Let $S(w)$ abbreviate $ w \cup \{w\} $, where $ w $ is some set (We can see that $\{w\}$ is a ...
5
votes
3answers
109 views
Gödel's incompleteness wrt weakend versions of ZFC
Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed.
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem
...
5
votes
3answers
68 views
Question Regarding the Axiom of Extensionality
Jech's text on Set Theory states the following:
If X and Y have the same elements, then X = Y :
∀u(u ∈ X ↔ u ∈ Y ) → X = Y.
The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
5
votes
5answers
228 views
Axiomatization of $\mathbb{Z}$
Though I've seen several cool axiomizations of $\mathbb{R}$, I've never seen any at all for $\mathbb{Z}$.
My initial guess was that $\mathbb{Z}$ would be a ordered ring which is "weakly" well-ordered ...
5
votes
4answers
151 views
Why would the author ask if I used the Associative Law to prove + is not equiv. to *?
I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows:
Problem 1: Show that + and * are necessarily different operations. That is, for any ...
5
votes
2answers
498 views
How can I write the Axiom of Specification as a sentence?
I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification".
To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...
5
votes
3answers
153 views
Pythagorean theorem and its cause
I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
5
votes
1answer
123 views
Axioms vs. Universal Constructions/Properties
What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...
5
votes
1answer
276 views
Can the power set be axiomatised?
I want to consider many-sorted first order logic with distinguished sorts $U$ and $P$.
Can I state a (finite?) set of first order formulae such that any model $M = (D^U, D^P, I)$ interprets the sort ...
4
votes
3answers
243 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 ...
4
votes
3answers
195 views
Existence and uniqueness of God [closed]
Over lunch, my math professor teasingly gave this argument
God by definition is perfect. Non-existence would be an imperfection, therefore God exists. Non-uniqueness would be an imperfection, ...
4
votes
5answers
532 views
Induction versus Natural Numbers
$0$ is finite. If $n$ is finite, then $n+1$ is finite. Hence, by induction, all numbers are finite. What is the catch?
4
votes
3answers
146 views
A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory
I have read in a book that the "axiom of foundation prevents anomalies such as a
set being an element of itself".
Now, axiom of foundation says that there exist an element in every set which is ...
4
votes
2answers
48 views
ZF Extensionality axiom
To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
4
votes
3answers
143 views
How does the axiom of regularity forbid self containing sets?
The axiom of regularity basically says that a set must be disjoint from at least one element.I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$ but it would seem to ...
4
votes
2answers
194 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.
