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Questions tagged [axioms]

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

37
votes
6answers
3k 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 ...
13
votes
2answers
1k 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.
37
votes
6answers
7k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
35
votes
8answers
3k 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 ...
112
votes
8answers
41k 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 Choice ...
20
votes
1answer
1k views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
9
votes
4answers
3k 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 ...
18
votes
3answers
4k 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 ...
20
votes
2answers
2k 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 ...
9
votes
2answers
2k views

What are the consequences if Axiom of Infinity is negated?

What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ...
7
votes
2answers
232 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
76
votes
7answers
5k views

Where are the axioms?

It is said that our current basis for mathematics are the ZFC-axioms. Question: Where are these axioms in our mathematics? When do we use them? I have now studied math for a year, and have yet to ...
42
votes
11answers
7k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...
23
votes
2answers
2k 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 ...
23
votes
2answers
6k views

Axiom of Regularity

I am having difficulty understanding Axiom of Regularity : Every non-empty set $\rm A$ contains an element $\rm B$ which is disjoint from $\rm A.$ So from my understanding if I have a set like: $...
13
votes
6answers
1k 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 ...
11
votes
1answer
316 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
9
votes
5answers
2k 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 ...
6
votes
1answer
384 views

When do surjections split in ZF? Two surjections imply bijection?

We have that the Axiom of Choice is equivalent to the principle that every surjection has a right inverse. However, without the Axiom of Choice we can determine for some $X$ that $X\succeq Y\implies Y\...
2
votes
3answers
1k views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
1
vote
2answers
290 views

Translating Tarski's Axiomatization/Logic of $\mathbb R$ to the Theory of Magnitudes

Update: This has become a project, but I need help. All answers will now be definitions, propositions, theorems, etc. that build on the theory. I will marks some of my own answers as community wiki so ...
-1
votes
2answers
302 views

Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?

According to 1 2, the third Kolmogorov axiom is for disjoint sets $(A_n)_{n \in \mathbb{N}}$ $P(\cup_n A_n) = \sum_n P(A_n)$ Is that really disjoint rather than pairwise disjoint? If we have ...
20
votes
3answers
2k views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
21
votes
4answers
2k views

Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
10
votes
3answers
635 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y (\...
18
votes
8answers
4k views

Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the "...
8
votes
1answer
786 views

Is there a general theory of the “improper” Lebesgue integral?

I like the Lebesgue integral a lot more than the other alternatives, because of its connections to measure theory; it's a way of thinking about integration that's as "liberated" from the structure of ...
9
votes
2answers
11k views

Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three ...
8
votes
1answer
2k views

Proving the pairing axiom from the rest of ZF

In ZF, the pairing axiom states that for every $x,y$ there exists the set $\{x, y\}$. Wikipedia also tells us we can dispense this axiom: This axiom is part of Z, but is redundant in ZF because it ...
19
votes
5answers
3k 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 ...
3
votes
2answers
2k 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 ...
3
votes
1answer
184 views

Axiom Schema of Separation Parameters

I'm doing some work in Jech's Set Theory book and I am a little confused about the definition of parameter in the Axiom Schema of Separation. Within a formula, what relationship would a parameter have ...
6
votes
1answer
330 views

A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at http://en.wikipedia.org/wiki/...
85
votes
10answers
17k 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?
79
votes
7answers
14k 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 "...
31
votes
4answers
3k views

Axiomatic definition of sin and cos?

I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ...
17
votes
2answers
3k views

What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?

Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, \...
9
votes
5answers
3k views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the (...
7
votes
9answers
1k 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
443 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 ...
7
votes
1answer
950 views

A model of geometry with the negation of Pasch’s axiom? [duplicate]

How would a geometry with all the usual axioms of Euclidean geometry, except that instead of Pasch’s axiom, we take it negation, look like?
3
votes
3answers
529 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
7
votes
3answers
2k 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
1answer
118 views

How should I interpret $y-x$ in this context?

I asked a related question here. I am meticulously working my way trough Mathematical Analysis by Apostol. I am reading about axioms, specifically axiom 4: Given any 2 real numbers $x$ and $y$, there ...
2
votes
1answer
250 views

Logical Conditional Truth Table Rationale

I am trying to get an intuition (ohh, the irony) about the logical truth tables. In particular, I am looking at the basic conditional P->Q with the following ...
5
votes
1answer
272 views

Does Euclidean geometry require a complete metric space?

Note: I'm not sure how to tag this question so please fix tags as appropriate and necessary. This question relates to Tarski's axioms of Euclidean geometry, not Hilbert's axioms. It says on the same ...
4
votes
4answers
319 views

Why is the infinite set from the axiom of infinity the natural numbers?

Why is the infinite set from the axiom of infinity the natural numbers? Is there any reason such set was chosen? Couldn't the axiom yield a set that looks like $\Bbb R$ for example?
3
votes
1answer
322 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
33
votes
8answers
4k 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 numbers,...
17
votes
3answers
4k views

Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...