# Aristotle's and Plato's even and odd numbers, sets and actual infinity [closed]

Plato and Aristotle both use the terms even and odd about numbers (and have a separate discussion of the number 1).

From this point, it seems, there would be no great distance to the sets of even and odd numbers. However, we know that the set-theoretical concept of the actual infinite is not acceptable to Aristotle.

Does this imply that even and odd are perhaps seen as properties of at least some numbers, not necessarily all, that even and odd are not to be looked upon as sets, and therefore that the sum of the two types of numbers not necessarily makes up all numbers - or that such summation perhaps would be forbidden?

• I suspect the type of question you're asking is not one that would make sense to Aristotle, who would probably fail to see the point of talking about the set of odd numbers and the set of even numbers --- this is a modern viewpoint. You may as well ask about the set of all mountains, rather than whether a certain mountain is less than 100 miles away or how many mountains someone has climbed. You don't need to contemplate the meaning of "set of odd numbers" to talk about the properties of numbers being odd. – Dave L Renfro Feb 27 at 18:44
• This is a sign of a truly revolutionary nature of Cantor's idea: after some time people ask, why this was not considered 2 or 3 thousand years ago:-) – Alexandre Eremenko Feb 27 at 20:40
• You might say that Aristotle was not too far from infinite sets, in that he addressed the actual infinite - and came out against it. After all the even numbers which he also addressed, are an example of a actual infinite set. – Mikael Jensen Feb 27 at 23:44
• Pythagoreans did not need actually or even potentially infinite sets to demonstrate that any given number (which to them meant positive integer greater than 1) was even or odd. That happened over a century before Plato and Aristotle. Aristotle's syllogisms already included statements like "all As are Bs", and in modern formalisms arithmetic can be done without any reference to sets as well. They are completely redundant for these simple types of propositions. – Conifold Feb 28 at 0:20
• Aristotle did consider actual and potential infinity explicitly, but not in connection with individual properties like even/odd, and not in the modern guise of sets. He rejected actual infinity, as did most philosophers and mathematicians after him, until Cantor, see How does actual infinity (of numbers or space) work? In modern terms, Aristotle's view corresponds to something like intuitionism. – Conifold Mar 1 at 7:18