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In the history of mathematics, are there notable examples of theorems which have been first considered axioms?

Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?

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    $\begingroup$ All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant. $\endgroup$
    – Asaf Karagila ♦
    Apr 8, 2019 at 13:16
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    $\begingroup$ I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems. $\endgroup$
    – quarague
    Apr 8, 2019 at 13:26
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    $\begingroup$ Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition. $\endgroup$
    – Asaf Karagila ♦
    Apr 8, 2019 at 14:38
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    $\begingroup$ And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD. $\endgroup$
    – Asaf Karagila ♦
    Apr 8, 2019 at 14:41
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    $\begingroup$ I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF? $\endgroup$
    – Bram28
    Apr 8, 2019 at 16:05

3 Answers 3

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The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms

In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.

http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf

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Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.

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Yes, everywhere. What is an axiom from one theory can be a theorem in another.

Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $\pi$ radians.

Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.

Also, watch this Feynman clip.

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  • $\begingroup$ That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms. $\endgroup$
    – Eyal Roth
    Apr 8, 2019 at 13:57
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    $\begingroup$ These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them. $\endgroup$
    – Ross Millikan
    Apr 8, 2019 at 20:03

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