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Background

When I was learning Math on Khanacademy, the teacher gave a brief demonstaration about the difference between 'theorem' and 'postulate/axiom'. He indicated that 'postulate' and 'axiom' means the same. I assume so in this post.

My understanding is following:

  • 'theorem' is a true statement established by means of assumptions.
  • 'postulate/axiom' is an assumption cosidered to be true (generally with high probability).

Question

Postulates/axioms used in a proof might turn out to be false. Does this mean it's possible that a theorem which has been proved can be false if one or more postulates/axioms are in fact false?

By the way, the following question is related, but I guess it doesn't answer my question directly. Once a mathematical theorem is proven true like the Halting problem can it ever be disproven?

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  • $\begingroup$ In math, axioms are not assumed to be "true" in any absolute sense. The axioms of a vector space are not necessarily true in a group, for example. They are just statements that one uses as a basis from which to prove other statements. If you try to use the axioms of Linear Algebra to prove statements in Group Theory, say, you may well wind up proving statements that do not hold for groups. $\endgroup$ Commented Jan 11, 2021 at 3:28
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    $\begingroup$ Axioms are used to define the system in which you are working. The axioms of vector spaces can't be false – they define what a vector space is. $\endgroup$ Commented Jan 11, 2021 at 3:30
  • $\begingroup$ Axioms need not be considered true, in fact maths with axioms that we consider false is perfectly fine, so long as it's not inconsistent. The extra condition that axioms should be considered true is relevant to applying maths in the real world, but in applied maths they don't care about rigor in the first place so why bother with axioms? $\endgroup$
    – SenZen
    Commented Jan 11, 2021 at 3:30
  • $\begingroup$ In arithmetic, one starting place is Peano's axioms which include that every number has a successor which is $1$ more than the previous number. If the axiom said only that each number has a successor that is $5$ more than the previous number, theorems based on them may still be true if no contradictions can be found. Of course there are consistent systems that contradict real world observations but the theorems can still be true in that they "prove" the relationships among the given "parts" including axioms. $\endgroup$
    – poetasis
    Commented Jan 11, 2021 at 3:31
  • $\begingroup$ One major advance in math came from assuming that parallel lines meet. There were no contradictions and the author [I forget his name] once said something like, "I have created an entire universe out of nothing." From this we got spherical geometry/trigonometry like the kind use with latitude and longitude on the surface of the Earth. $\endgroup$
    – poetasis
    Commented Jan 11, 2021 at 3:35

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