# What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says:

In practice, most mathematicians ... do not work from axiomatic systems

Is this correct? If so, what is an example of this?

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## closed as off-topic by Ryan Budney, Daniel Moskovich, Qiaochu Yuan, Steven Landsburg, Nik WeaverFeb 12 '14 at 3:35

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Daniel Moskovich, Qiaochu Yuan, Nik Weaver
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I believe this just means that most mathematicians, most of the time, do not go out of their way to make explicit exactly what axioms (e.g., choice) they are using; and also do not write complete (e.g. two-column or semantic tableaux) proofs in practice. I'm not sure this question is appropriate for MO. – Noah Schweber Feb 12 '14 at 2:50
An example would be almost any mathematics paper in the literature. – Ryan Budney Feb 12 '14 at 2:51
Yes, I think the sentence is unclearly written. Surely the intent is to say that mathematicians commonly do mathematics without paying much if any attention to foundational issues. (Not that group theorists don't use the axioms of group theory, or that topologists don't use the axioms of topological spaces.) – Tom Goodwillie Feb 12 '14 at 3:01
A key point - drawing out what Tom Goodwhillie said - is that most results in group theory are actually proved in some stronger theory (for example, the axioms of group theory do not include any axioms that would allow for the construction of a product group from two given groups). The stronger theory in which day-to-day mathematics is carried out is not formalized, although of course it can be. But I suspect few mathematians can state all the axioms of ZFC, for example, which casts doubt on the claim that the are really working in ZFC in their informal mathematics. – Carl Mummert Feb 12 '14 at 13:44

Most mathematics is not done in axiomatic systems. As a trivial example, consider a typical MathOverflow post: it is in English, is not in a formal language, and does not specify an axiom system. The more important claim is different: $\$ Most mathematics can be readily formalized in axiomatic systems, and it's sometimes useful to do so.