How to find a 'set' after knowing the axioms of ZFC?

Question

This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are set, behave with some other operations defined on the set.

My question is how exactly do we find objects do fulfill these axioms? Is there some formal procedure for it , or, is it just guess work?

Answer 1

No, the axioms of ZFC absolutely do not tell us how to "check if something is a set". It seems like you are imagining a situation where you have a mathematical object and then you somehow use the axioms of ZFC to check if this object is a set. No such situation exists.$^*$

You seem to be conflating the status of the axioms of a theory like ZFC with axioms defining, say, a vector space. In the case of vector spaces, you are given a set and two operations which are claimed to be scalar multiplication and vector addition. To check if this is a vector space, you check whether the axioms defining vector spaces are satisfied by these operations.

By contrast, the axioms of ZFC are simply a list, not necessarily exhaustive, of statements we hold to be true about sets. Most of them tell us that a certain set exists if some other sets exist, such as the powerset of a set, or the union of a set of sets, or a subset of a set defined by some formula. If you want to "find objects [which] fulfill these axioms", you simply start from sets which are postulated to exist and then apply these constructions.

For example, from the set of integers, one construct the set of pairs of integers (using the axiom of pairing), viewed as formal fractions. To obtain the rationals, one restricts to a certain subset of this set (using the axiom of specification), say the set of coprime pairs of non-zero integers where the second element is always positive plus the pair $\langle 0, 0 \rangle$. To obtain the reals, one constructs the powerset of the rationals (using the powerset axiom) and restricts to certain special sets of rationals corresponding to Dedekind cuts (using the axiom of specification).

$^*$ "What about when we show that the set of all sets does not exist in ZFC?" That's different from showing that something is or is not a vector space. The "set of all sets" is not a validly specified mathematical object. You're not first describing a legitimate mathematical object and then measuring it against the axioms of ZFC to show that it is not a set. You're simply showing that no set satisfying a certain description (namely, that it contains every set) exists.

Answer 2

Remember when you started learning calculus with some rigor? At some point you were introduced to the idea of limits. Exactly which limit concepts you were introduced to first differs from person to person, but one way might have been to first define the limit of a function $f(x)$ at some point $a$:

• $\lim_{x \to a} f(x) = L$ if $\forall \epsilon > 0, \exists \delta > 0, 0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$.

This definition is then used to prove a bunch of properties of limits.

But then you were introduced to limits at infinity, which needed two more definitions:

• $\lim_{x \to \infty} f(x) = L$ if $\forall \epsilon > 0, \exists M, x > M \implies |f(x) - L| < \epsilon$,
• $\lim_{x \to \infty} f(x) = L$ if $\forall \epsilon > 0, \exists M, x < M \implies |f(x) - L| < \epsilon$.

And these definitions were then used to prove pretty much the same properties before, except for the new limits.

And then you were introduced to one-sided limits:

• $\lim_{x \to a+} f(x) = L$ if $\forall \epsilon > 0, \exists \delta > 0, a < x < a+\delta \implies |f(x) - L| < \epsilon$.
• $\lim_{x \to a-} f(x) = L$ if $\forall \epsilon > 0, \exists \delta > 0, a -\delta < x < a \implies |f(x) - L| < \epsilon$.

And these definitions were used to prove the same bunch of properties also hold for one-sided limits.

And then you were introduced to infinite limits, where the $L$ is $\pm\infty$, which duplicates each of the above definitions twice more, with the requisite changes, and are then used to prove that almost the same bunch of properties also hold for them.

And then there are sequences ...

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There comes a point when mathematicians get tired of proving pretty much the same things with only minor variations over and over again. This is when they invent a new theory. They look at all the examples they've been running into, and try to figure out a common scheme that they all can fit into:

$\lim_{x \to a} f(x) = L$ if for every neighborhood $V$ of $L$, there is a neighborhood $U$ of $a$ such that if $x \ne a$ and $x \in U$, then $f(x) \in V$.

All the previous definitions of limits fit the pattern with different definitions of "neighborhood".

• For basic limits, a neighborhood of $a$ is a set $(a-\epsilon, a + \epsilon)$ for some $\epsilon$.
• For limits at infinity and infinite limits, neighborhoods of $\infty$ and $-\infty$ are $(M, \infty)$ and $(-\infty, M)$ for some $M$.
• For one-sided limits, neighborhoods $a$ are $[a,a+\epsilon)$ or $(a-\epsilon, a]$, depending on which side you are after.

To prove the properties of limits we want, we need some conditions on what counts as a "neighborhood":

• $a$ should be in any neighborhood of $a$ (if $a$ actually exists, and is not a shorthand for a more complicated concept, as with $\pm\infty$).
• If $a \ne b$, then there should be neighborhoods of $a$ and $b$ that do not intersect.
• If $U$ and $V$ are both neighborhoods of $a$, then there should be another neighborhood $W$ of $a$ with $W \subset U \cap V$.

From these properties, more generalization occurs - for instance, maybe neighborhoods of $a$ and $b$ could be allowed to intersect, until one finally arrives at the theory of Topology.

In this new theory of topology, one can now prove a lot of results just from the axioms (which of course changed markedly from those beginnings). All of these proofs no longer had to be redone in every space, or every topology. All that is needed now to have all those powerful concepts such as continuity and connectedness and compactness, and all those powerful theorems are available, once you confirm that some particular object you are trying to study satisfies those axioms of topology.

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That is the mistake you are making. These theories were not just pulled out of someone's arse, and then they started looking for objects that satisfied the axioms. No. They started with the objects, and devised the theory to fit their needs.

And when you are learning them, it is still not your purpose to figure out objects that fit the axioms - though that can be a useful exercise to make sure you understand those axioms. You will be given some examples, usually immediately after the axioms are laid out, if not before. Your purpose is to figure out what can be proved from them.

And later on, when your studies happen to bring up some object, you can then check if that object meets the axioms of the theory. And if so, you can apply all the results of that theory to your object without having to redevelop the whole thing in this one special case.

The objects came first in the invention of the theory. The objects will also come first in any application of the theory. And when you are developing the theory, you don't want to work with some specific object, because if you do, your results will apply to that specific object, and not the theory as a whole. As such, when you are studying some other object that meets the axioms in the future, those results would not necessarily apply to it.