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

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3
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2answers
220 views

axioms of equality

Every text I've come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other's elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in ...
3
votes
2answers
364 views

Why is matrix multiplication defined a certain way? [duplicate]

Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined ...
3
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2answers
2k 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 ...
2
votes
2answers
81 views

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
2
votes
2answers
543 views

What are the primitive notions of real analysis?

My dad introduced my to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition? ...
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0answers
60 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 - ...
1
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1answer
133 views

What makes Tarski Grothendieck set theory non-empty?

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
1
vote
1answer
150 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 ...
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6answers
659 views

Is it possible to have a field without an additive identity?

If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization? By definition it is not possible, but are there ...
0
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0answers
30 views

Verifying certain congruence axioms in taxicab geometry

Given: I need some help I've shown what I have so far d(A, B) = |a1 − b1| + |a2 − b2| where A = (a1, a2) and B = (b1, b2). Some people call this ...
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votes
1answer
26 views

Axiom of infinity: What is an inductive set?

This Axiom states that there exists an inductive set. But, what is the definition of an inductive set?