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I thought I understand what is a set long ago, that is, a collection $S$ of some stuff satisfying some "well defined" rule so that either $a\in S$ or $a\notin S$. But recently I became less sure.

For example, consider subsets of integers. I thought it is something like $S=\{n\in{\mathbb Z}\,|\, property\}$, where "property" is some "rule" to decide if $a\in S$ or not, like " $n$ is an even number that cannot be written as the sum of two primes". However, each "rule" can be formalized into an English paragraph so that it uniquely decides the rule, but the set of English paragraphs is countable, this means there are at most countably many "rules". Now there are uncountably many subsets of ${\mathbb Z}$. So there are subsets in ${\mathbb Z}$ without "rules", I guess meaning only God knows whether $n\in S$ or not, or even He does not know? It is disturbing to think like this, especially when we often use languages like "let $S$ be an arbitrary subset of ${\mathbb Z}$".

I am aware that there is no formal definition of set. I would like to hear any comments on how to think about what is a set, hopefully regain some peace in mind.

Yuval
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    Have you tried searching the site? For example, https://math.stackexchange.com/questions/1452425/what-is-the-definition-of-a-set or https://math.stackexchange.com/questions/3717628/can-anyone-please-help-me-to-understand-what-does-well-defined-mean-in-the-def or even https://math.stackexchange.com/questions/4242526/on-the-definition-of-well-defined-object may already contain answers to your question... – Asaf Karagila Mar 28 '22 at 20:38
  • Don't worry about running out of numbers. You can do an amazing about mathematics naming only the single number 0, and simply infer the existence of the others without actually naming them. – Dan Christensen Mar 29 '22 at 00:26
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    Welcome to model theory. Once you start asking these questions, things get weird fast. Sets are these funny mathematical things we invented to model reality. Most of the sets we care about can be expressed using a finite number of symbols. There are other sets out there, and lots of things might be sets but might not be, depending on the model we happen to be operating in. ZFC effectively starts with the empty set and says "all of these nice things are sets", and then (ax. of foundation) declares that certain things are not sets. All of the things in between are left intentionally ambiguous. – Silvio Mayolo Mar 29 '22 at 04:46
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    Very nice observation. I think the right way to deal with such questions, is to either ignore them (and just stick to 'the rules of set theory')/ponder them on a sunday afternoon, or spend your career on it and fully devote yourself to the foundations. Most likely there is nothing to be won at working on the foundations, but it is with art: the artist needs to do the art. If you feel like you need to ponder these question go ahead. Or relax and live with it that math is like venice (necessarily) build on a shaky foundation. – lalala Mar 29 '22 at 08:01
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    Does it bother you similarly that there are uncountably many real numbers, but that you could only ever assign descriptions to countably many of them? – preferred_anon Mar 29 '22 at 08:29
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    @preferred_anon It is consistent with ZFC that we can assign a unique description to each and every single real number, indeed to each and every single set in the universe. – Asaf Karagila Mar 29 '22 at 16:36
  • @AsafKaragila I must be missing something - I mean finite descriptions, taken from a finite alphabet. Aren't there only countably many of those? – preferred_anon Mar 29 '22 at 16:51
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    @preferred_anon: Yes, but "the set $x$ is definable" is not a first-order property. It is a property of the meta-theory. Look up "pointwise definable model" in the context of set theory, and there had been some good discussions on this on this website as well. – Asaf Karagila Mar 29 '22 at 16:53

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I find the reason you lost your piece of mind is rather problematic... Applying your reasoning to the real numbers we can reach the same "conclusion", namely there are real numbers that "only God knows" what they are. The example of the reals can even become equivalent with the subsets of $ \mathbb{Z} $, if we order $ \mathbb{Z} $ then every subset of $ \mathbb{Z} $ can be described by a sequence of zeros and ones (its characteristic function) - which can be thought of as the binary representation of a real number in $ [0,1] $. So every subset of $ \mathbb{Z} $ is now merely a point on an interval (there are some minor technical problems with this mapping but they don't affect the essence of the argument)...

Sets can be really mind-bending (axiom of choice, non-measurable sets...) and it is a fact that Cantor wrote to Dedekind "I see it, but don't believe it!" on a set theory question but these difficulties don't have to do with how and if we can "express" every single set in writing.

PS the set of English paragraphs may be countable, but the set of English paragraphs that can be read by a human being during a period of 20.000 years is actually finite - so until we figure out a way to live forever we are bounded as a species by finiteness

shortmanikos
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What you say is kind of true, but you're talking about a different thing. Sets are well defined, Set Theory is built from what we call the Zermelo-Fraenkel Axioms. The thing is that, as you have correctly pointed out, we can't "define" every set through given rules. What you describe has more to do with what we call partially recursive sets, which are sets such that you can verify in human terms if an element is or is not there. The definition of set has problems, but they are of other nature. And the fact we can't construct certain sets need not mean we can't work with them, sets we can describe are just particular examples of sets, and you might think they cover all just because we can't "see" any other kind of set

vertex
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Evaristo
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The very concept of sets exists to avoid your worries.

We can formalize properties using a programming language. Specify some basic operations you are allowed to perform on objects (depending on context, these could be set inclusion, addition, …). Then any syntactically correct use of these operations is a so-called "atomic" property; any finite combination of atomic properties using "and (${\wedge}$)" and "or (${\vee}$)" is also a property. If you want, you can also use infinite combinations of atomic properties too (but see below), or allow yourself to introduce "local variables" using the existential (${\exists}$) and universal (${\forall}$) quantifiers.

For example, allow yourself to use addition, multiplication, and quantifiers, and assume all values are integers; then one can immediately write down a property to test for "is prime": $$\text{prime}(p)=(\forall a,b)((ab\neq p)\vee(a\cdot a=a)\vee(b\cdot b=b))$$ "$n$ is an even number that cannot be written as the sum of two primes" can then be written as $$\phi(n)=(\exists q)(q+q=n)\wedge(\forall p,q)(\neg\text{prime}(p)\vee\neg\text{prime}(q)\vee p^2+q^2\neq n)$$

But the programming-based approach leads to trouble. Suppose $\phi$ is a property. Then any proposition about $S=\{x:\phi(x)\}$ is implicitly a proposition about $\phi$:

  • "$x\in S$" is the same as "$\phi(x)$"
  • "$S$ is uncountable" is the same as "uncountably many points satisfy $\phi$"
  • "$S$ is closed" is the same as "$\phi$ is preserved under limit operations"
  • etc.

So why bother with sets at all? Because, as you fear, some properties are hard to write down. Instead, we try to construct a system which doesn't require them to be written down at all.

A set is defined as an object $O$, such that, for any other object $x$, either $x\in O$ is true, or it is false. One way to decide which of these propositions holds is to use a rule: $$x\in O\Leftrightarrow \phi_O(x)$$ where $\phi_O$ is some property. But a rule is not required.

If your "programming language" is sufficiently rich, using sets doesn't actually save you anything; one can define $\phi_O(x)=(x\in O)$. Even if you don't allow use of ${\in}$, you still can just write down all the elements and call that a "property": $$\phi_O(x)=\left(\bigvee_{y\in O}{y=x}\right)$$ But for this reason, most mathematical logicians try to use programming languages that allow neither trick, and then one can prove via an elementary cardinality argument that there must exist sets without a corresponding property.

Jacob Manaker
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