# Tagged Questions

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

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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $X\succeq Y\implies Y\... 7answers 2k views ### Is$\{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$. This ring satisfies the field axioms: Closure under addition. ... 2answers 933 views ### What are the consequences if Axiom of Infinity is negated? What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ... 2answers 946 views ### what is the relationship between ZFC and first-order logic? In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the ... 2answers 155 views ### Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness? According to 1 2, the third Kolmogorov axiom is for disjoint sets$(A_n)_{n \in \mathbb{N}}P(\cup_n A_n) = \sum_n P(A_n)$Is that really disjoint rather than pairwise disjoint? If we ... 1answer 798 views ### Can we prove the existence of$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, ... 9answers 926 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 ... 2answers 379 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 ... 5answers 701 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 ... 3answers 370 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$3$... 2answers 1k 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$Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$.... 1answer 233 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/... 1answer 84 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$y$, ... 10answers 15k 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? 6answers 8k 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 "... 4answers 2k 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 ... 6answers 2k 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 ... 3answers 1k 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 ... 6answers 2k 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,... 3answers 359 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 (\... 1answer 1k 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 ... 4answers 1k 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 ... 2answers 249 views ### Is$\mathrm{ZFC}^E$outright inconsistent? From$\mathrm{ZFC},$define a new theory$\mathrm{ZFC}^E$by adjoining a constant symbol$E$together with axioms to the effect that:$E$is countable and transitive$(E,\in)$is an elementarily ... 5answers 716 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 ... 2answers 375 views ### Non-constructive axiom of infinity Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ... 4answers 1k 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 (... 2answers 146 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 ... 3answers 830 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 ... 1answer 403 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 ... 1answer 468 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 ... 3answers 1k views ### Defining the Complex numbers I posted this question nearly 10 days ago, but am still really not satisfied with the answers I got, I have no prior education in abstract algebra, group theory, or other abstractions, and most of the ... 4answers 305 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 ... 2answers 119 views ### Is my proof of the uniqueness of$0$correct? I am working my way through "Mathematical Analysis" by Apostol. What I am attempting to prove is that if there exist$q_{1}$and$q_{2}$such that$x + q_1 = x$and$y+q_2=y$, then$q_1=q_2$... 1answer 78 views ### The concept of negative numbers in the$4^{th}$field axiom I just started working my way trough "Mathematical Analysis",$2^{nd}$Edition by Apostol. I am reading every detail very carefully to try to get a rigorous understanding. The$4^{th}$axiom in that ... 1answer 351 views ### Prove$ L^2$inner product satisfies positivity There is a proof in my textbook where I am a little bit unsure about a small detail. It would be great if someone could clarify it for me. We are supposed to prove positivity of the$L^2$inner ... 6answers 284 views ### How do we define arc length? In trying to write a nice proof of the derivatives of$\sin(x)$and$\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ... 2answers 109 views ### ¿Can you help me with axiom of regularity? [duplicate] I understand that if we define a set such that$A=\{A\}$we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I ... 2answers 101 views ### Proof using only field axioms Prove if$x, y ∈ R$and$xy > 0$then either$x > 0$and$y > 0$, or,$x < 0$and$y < 0$using only the field axioms. These include the Field axioms for addition, multiplication, ... 1answer 74 views ### Whitehead's axioms of projective geometry and a vector space over a field According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ... 9answers 6k views ### Does mathematics require axioms? I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ... 11answers 3k views ### What is exactly the difference between a definition and an axiom? I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that$\forall n:0\...
Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...