As sets are the building blocks of modern mathematics, I would think that every mathematical object can be represented with sets. First, is it true? Second, if it is true, how tuples can be represented without the use of tuples (ordered pairs)? I've seen a representation using functions, but functions are represented with a set of ordered pairs.
Asked
Active
Viewed 191 times
1
-
2Yes, virtually all of pure mathematics can be interpreted in set theory. The most common definition of the ordered pair $(a,b)$ is the set ${{a},{a,b}}$ (this definition is due to Kuratowski). It can be proven from this definition that $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. Any other definition which has this property would work, so some authors do give the ordered pair $(a,b)$ a slightly different set-theoretic definition. – Joe Jan 30 '24 at 17:01
-
Thank you for your answer! Could you post it as an answer of my post so that I can put it as answered? – Nathan Kaufmann Jan 30 '24 at 17:04
-
I'm glad I could help. I am sure your question has been asked before, so I did not post my comment as an answer. I am currently trying to find a suitable duplicate target. In the meantime, if you want to delve deeper into the "philosophy" behind implementing mathematics in set theory, I would highly suggest reading this answer by Andreas Blass on MathOverflow. – Joe Jan 30 '24 at 17:12
-
If virtually all of pure mathematics can be interpreted in set theory, how can a class be interpreted in set theory? – Nathan Kaufmann Feb 06 '24 at 19:56
-
The objects that the axioms of $\mathsf{ZFC}$ are intended to describe are sets, and so you make a valid objection there. However, there are ways to get around this issue: one way to do this is to use a theory that does allow for both sets and classes, such as $\mathsf{NBG}$. Another way is to think of classes not as objects in their own right, but as syntactic formulas. I am sure that there is much discussion of this on Math.SE and in a number of set theory textbooks. – Joe Feb 06 '24 at 20:15