For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.

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6
votes
5answers
670 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: ...
3
votes
0answers
46 views

A mathematical object that can be characterised by axioms as well as by universal constructions

This question (more specifically one of the comments) prompted me to ask this question: Can someone give me an example of a mathematical object that one can characterize both via axioms and universal ...
7
votes
5answers
353 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
191 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
2answers
326 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.
3
votes
2answers
128 views

Why does any transitive model satisfy extensionality?

I see it stated as something very clear, but i can't figure it out. i found a proof in Jech(old version) which goes through the concept of restricted formulas, which i don't quite understand. (i'm not ...
3
votes
1answer
426 views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
2
votes
2answers
108 views

What percentage of formulas is unprovable in a given axiomatic system?

I am trying to use language I am not familiar with, so bear with me. If I make no sense, I try to be clearer. Assume we are given a formal language. Assume $S$ is the set of every well-formed formula ...
3
votes
1answer
169 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 ...
2
votes
2answers
132 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
1
vote
2answers
178 views

How can we know arithmetical axioms are consistent?

If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal. ...
2
votes
5answers
214 views

Is there an axiom that prevents other axioms from contradicting each other?

i.e. Does an axiom already exist, which prevents the addition of those new axioms which can contradict already existing axioms? Also, who decides that something is an axiom?
2
votes
2answers
201 views

Is all mathematics based on the concept that $1+1=2$?

Thought about this recently, and was a bit stuck. Is all mathematics based on the concept that $1+1=2$? For example, if $1+1\ne2$, then all arithmetic won't work, right?
-1
votes
1answer
98 views

Arbitrary definition [closed]

I find it most perplexing for how the definition of say, a set, defines itself. How does it become tangible? Where does it come from? Because, arbitrary definition seems more so like just saying "it ...
3
votes
2answers
141 views

How bad is this analogy for logical independence?

It is an amazing and well-known fact that the Continuum Hypothesis is logically independent of Zermelo-Frankel set theory with the Axiom of Choice (ZFC), assuming it is consistent. In a similar vein, ...
0
votes
1answer
47 views

Are there any set of rules on when induction can or cannot be applied?

I know this is some sort of "common-sense" question, but I want to get a clear boundary on this: when can I apply / cannot apply induction on a proof? For example, I know that: Ex1) A person with ...
2
votes
1answer
98 views

Can we apply a successor operation in Peano's arithmetic infinitely many times, is the successor well defined?

If we look at the axioms of Peano arithmetic, e.g. http://mathworld.wolfram.com/PeanosAxioms.html, they contain an axiom: If $a$ is a number, the successor of $a$ is a number. However, the axioms do ...
0
votes
2answers
129 views

relationship between Zorn's lemma and Axiom of Completeness

For me , they look like they are 'similar' to each other , just that one is used in set and another one is used in numbers. Can anyone tell me is there any relationship between Zorn's Lemma and Axiom ...
6
votes
3answers
499 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
3answers
330 views

Why is axiomatic system needed in propositional logic?

I am trying to learn propositional logic. I have read that axiomatic system is defined since there are some problems which can not be solved using truth tables. I have found such a problem in ...
9
votes
3answers
292 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 ...
0
votes
1answer
106 views

A substitution inference in predicate calculus

I'm not sure what I'm missing here, but I have this lemma I am trying to prove, and it is giving me a lot of trouble. I'm technically working in ZF set theory, but this part doesn't need much more ...
4
votes
3answers
402 views

What properties are allowed in comprehension axiom of ZFC?

I am trying to understand the axioms of ZFC. The axiom schema of specification or the comprehension axioms says: If A is a set and $\phi(x)$ formalizes a property of sets then there exists a set C ...
3
votes
3answers
205 views

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these ...
2
votes
2answers
140 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 ...
8
votes
3answers
462 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 ...
21
votes
6answers
822 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). ...
14
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, ...
3
votes
2answers
646 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 ...
1
vote
3answers
378 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 ...
1
vote
1answer
55 views

In class universe, on what basis can a conclusion that every element of a set is set be made?

Is every element in a set set? In a set model it is obvious true. However in the class universe it is another story since it need to be shown not a proper class. Let $A$ be a set, $a \in A$. ...
5
votes
0answers
79 views

Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
1
vote
0answers
143 views

Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
2
votes
2answers
240 views

Why do we need the axiom of completeness? Why won't Cantor's diagonalization work without it?

Okay so for my upcoming test I need to "be able to explain at least one result that would not hold if the axiom of completeness were not accepted" My teacher suggested that I could try to explain why ...
2
votes
2answers
750 views

Is Gödel's theorem invalid? [closed]

Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available). As it seems very plausible, I ask for any references and scrutinizations of the paper.
1
vote
1answer
77 views

What will happen when Ax.Inf is replaced to its negation in ZFC? [duplicate]

Possible Duplicate: What are the consequences if Axiom of Infinity is negated? In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable? ...
10
votes
2answers
517 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 ...
1
vote
3answers
45 views

Is this definition true about comparisons of sets?

I'm trying to find the error in a proof that yields a contradictory result, and I'm beginning to think that one of the definitions I start with is incorrect or self-contradictory. Is the following ...
4
votes
3answers
318 views

Please help. Great confusion about the "two levels of discourse'' mathematical logic.

I regard mathematics as being build up in the following way: We have some collections of symbol and rules (which are and have to be described in a natural language) to manipulate these symboles. If we ...
3
votes
1answer
620 views

Proving $1 > 0$ using only the field axioms and order axioms

How do I prove $1 > 0$ using only field axioms and order axioms? I have tried using the cancellation law, with the identities in a field and I cannot get anywhere. Does anybody have any ...
2
votes
1answer
197 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 ...
8
votes
2answers
246 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 ...
1
vote
1answer
144 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 ...
-1
votes
2answers
458 views

Axioms of set theory and logic

Zermelo–Fraenkel set theory is the most common foundation of mathematics with eight axioms and axiom of choice (ZFC): http://plato.stanford.edu/entries/set-theory/ZF.html But one can see that the ...
3
votes
1answer
225 views

Are there ways to describe the Martin Axiom intuitively?

I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking ...
40
votes
9answers
12k 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?
2
votes
1answer
105 views

Is failing to admit an axiom equivalent to proof when the axiom is false?

Often, mathematicians wish to develop proofs without admitting certain axioms (e.g. the axiom of choice). If a statement can be proven without admitting that axiom, does that mean the statement is ...
5
votes
2answers
285 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 $1$. ( ...
2
votes
1answer
203 views

How do I prove the existence of infinite union in ZFC?

Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?
3
votes
2answers
94 views

Does the negative axiom hold in a universe containing just 0 and 1?

I am self-studying the wonderful book, Elementary Geometry from an Advanced Standpoint. In chapter 1, problem 19 it says: Suppose that the elements of R were 0 and ...