Synthetic Axiomatization - Meaning

Question

I was reading an answer about the construction of the real numbers and there was a mathematical therm that I could not understand.

On that answer they said that the real numbers can be constructed through synthetic axiomatization, what is the meaning of this expression in mathematics?

Thank you for your time

score 0 · Accepted Answer · answered Jan 17 '21 at 14:05

0

Whatever you read was wrong. You cannot construct something via any kind of axiomatization. What you can do is to stipulate that the real numbers are a collection of objects together with some operations on them such that as a whole they satisfy some axiomatization. Of course, when you do so you cannot assume that there is such a thing at all! That is why it is a non-trivial fact that there is in fact a model of the standard axiomatization of the reals. See also this post regarding the two common kinds of axiomatizations of the real numbers (the standard one is the second-order one).

answered Jan 17 '21 at 14:05

user21820

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Thank you, your answer and the link were very useful – Beaba Jan 17 '21 at 17:37
@Beaba: You're welcome! If you feel that my answer has addressed your question, you can accept it by clicking on the tick. – user21820 Jan 17 '21 at 18:01

Synthetic Axiomatization - Meaning

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