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

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Automatic theorem prover for proving simple theorems?

Is there a simple software that I could use to practice proving theorems in my course of mathematical logic? Basically what I need is ability to 1) define what axioms and laws I am allowed to use in ...
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2answers
37 views

Let $F$ be a field and $x, y\in F$. Prove:

Use field axioms to prove: a) $(−1) · (−x) = x $ b) If $x · y = 0$ then $x = 0$ or $y = 0$ I don't understand how to approach these questions. Does the field include $1$ and $0$ as well?
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1answer
45 views

Is is possible to prove that two systems of axioms are equivalent

The question is self explanatory. Is it possible to prove that two systems of axioms in two very different branches of math are equivalent? Is there a textbook to help me? Thanks!!
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1answer
32 views

Alternative Axiomatic Systems

At least as I understand the motivation behind rigorous definitions of the foundations of mathematics (the only contender with which I'm familiar being ZFC and extensions), the idea of an axiomatic ...
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1answer
24 views

Writing the ZF formula for the choice function (given Well-ordering)

If every set has a well order, then the axiom of choice follows: Given a well order on $\bigcup_{i \in I} A_i$ we define the choice function in this way $f(i) =$ "the first element of $\bigcup_{i \in ...
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1answer
75 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
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1answer
38 views

Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha $ is an ordinal and $ x \in V_\alpha)$ I believe I understand how to ...
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1answer
58 views

How to know when a system of axioms is 'complete'?

Here, I (basically) stated the group axioms as follows. $(xy)z=x(yz)$ $xe=x, ex=x$ $xx^{-1}=e$ In that post, answerers Martin and Ittay were critical of the above list for not including ...
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1answer
86 views

Real numbers axiomatization without natural numbers

I think to remember that there is way to uniquely characterize the real numbers $\mathbb{R}$ via an axiom set. I wonder if this is possibly without introducing some notion of the natural numbers ...
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1answer
46 views

Basic Field Properties: multiplication

I am struggling with the proofs: a) $(a^{-1})^{-1} = a$ b) $(-a)^{-1} = -a^{-1}$ I have done the rest of the theorem but it is just these two that are difficult. To prove them you can only use the ...
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0answers
59 views

Axiomatizing topology through continuous maps

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 $Y$ ...
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0answers
161 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
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0answers
86 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) ...
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0answers
46 views

Strength of “Every finite dimensional subspace of a vector space has a complement”

Does the following choice principle have a name? Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary ...
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0answers
68 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
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0answers
134 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
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0answers
46 views

Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
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0answers
48 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 ...
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0answers
37 views

Are there any formal theories that are not axiomatic?

Are there any formal theories that are not axiomatic? Could you give me some examples?
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0answers
63 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
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0answers
113 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
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0answers
41 views

Is “autonym” an autonym?

I believe this is set theory, but I don't really know set theory, so... An autonym is a word that refers to a property that the word itself happens to possess. For example "noun" is an autonym ...
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0answers
57 views

Congruent figures have equal areas. Why?

When we are first introduced to the idea of congruent polygons (generally triangles) in school, we often define congruent figures as "figures which have the same shape and size". However, today, ...
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0answers
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How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
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0answers
36 views

Set of true statements generated by set of axioms with a binary operator

I wondered about this and I am having a hard time formulating it as a question at all, but I hope I can express something if my wondering here. Assume we have a set $V = \{\mathbb N, +, =, X ,(,)\}$. ...
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0answers
64 views

Hilbert projection theorem without countable choice

All the proofs of the Hilbert projection theorem, existence part, that I have seen so far use countable choice (usually implicitly). Is this necessary? It seems like you might be able to leverage the ...
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0answers
28 views

Transitive property of equality and the fundamental nature of algrebra

The fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we ...
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0answers
70 views

Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
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0answers
58 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 - ...
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0answers
144 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 ...
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0answers
32 views

Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...
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23 views

How to translate these defitions of bipartiteness to each other?

Let $(V,E)$ be a bipartite graph. I'm trying to capture this property and I've come up with two definitions and am surprised/confused that one uses a negation and the other doesn't - still I can't ...
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0answers
61 views

Proof of congruent angles from Hilbert's axioms

I'm looking for a proof, using Hilbert's plane axioms (compiled, for instance, here), of the congruence of the four blue angles. where the lines $CE$ and $HF$ are parallel. This is a well known ...
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0answers
186 views

How are multiplication and addition defined?

Many people argue that multiplication is not necessarily repeated addition, indeed in the field of order 4 given on Wikipedia, adding one to itself A times will never yield A*1. See: ...