0

Here's what happens: most mathematicians study the basic foundations of set theory and boolean logic in one course, and then never ever think about those foundations ever again... that is, until something goes wrong in mathematics such as needing to define the set of equivalence classes as the elements of an $L_p$ space rather than functions themselves, or perhaps for instance the controversy over Riemann-zeta regularization only yielding logically consistent results when the axiomatic definitions of convergence are changed to something other than their conventionally accepted definitions.

I don't want to make that mistake, so I want to be more sure of my statements in a mathematical setting by knowing they are logically consistent starting with something elementary.

Let's say $X$ and $Y$ are two metric spaces as defined here https://en.wikipedia.org/wiki/Metric_space .

Now, consider two elements of the metric space $X$ denoted $a$ and $b$, $a,b \in X$.

Then, consider a function from the set $X$ to the set $Y$ denoted $f,$ where $f(x)$ denotes the action of $f$ on some element $x \in X.$

Then, how can I use set theory and the definition of an equivalence relation as well as boolean logic prove in a logical way that

$a = b$ implies $f(a) = f(b)$?

CheeseBlues
  • 101
  • 2
    "$a=b$ implies f(a)=f(b)" is just the substitution property of equality (or the definition of "function", depending on how you look at it). And what does the definition of equivalence relation have to do with anything? – spaceisdarkgreen Sep 12 '23 at 04:16
  • 4
    Topology is irrelevant as well. – Anne Bauval Sep 12 '23 at 04:17
  • Thank you for the comments. So, some people take this as an axiom of the equality relation apparently. But, if you don't, is there then a way to prove the substitution property otherwise? – CheeseBlues Sep 12 '23 at 04:20
  • If it looks trivial to you, check out how much work it takes to prove $2 + 2 = 4$: https://math.stackexchange.com/questions/2860346/how-to-prove-22-4-using-zermelo-fraenkel-set-theory?rq=1 . This is why I want to be more sure, because I know these statements are more complicated underneath than the for-granted way I most often use them. – CheeseBlues Sep 12 '23 at 04:24
  • 1
    @CheeseBlues Lookup First-order logic with equality, also 1, 2, 3 and links therein. – dxiv Sep 12 '23 at 04:30
  • 6
    #CheeseBlues: if you'd like to see a ZFC proof that for all $A,B$ sets, if $$ is a function $A\rightarrow B$, then for all $x,y\in A$, $f(x)=f(y)$, then I suggest cutting down on the flavor text, removing the irrelevant details about metric spaces, and asking for this explicitly. If you.re after something else, you'll probably need to clarify your question, as it's hard to tell right now. – Z. A. K. Sep 12 '23 at 04:32
  • 2
    @spaceisdarkgreen: Technically, this is npt the substitution property of equality: that would apply if $f$ was a function symbol in the language of ZFC. This $f$ is not a function symbol (there are no function symbols in the language) but an ordinary bound variable satisfying some conditions. Defining what is even meant by the notation $f(x)$, and proving that the implication actually holds, has at least a tiny bit of non-trivial content. – Z. A. K. Sep 12 '23 at 04:38
  • 2
    @Z.A.K. Yeah, thus the "or the definition of function..." part. My intention was more to cut through the onslaught of irrelevancy and point out that it was just a logic/basic definitions thing than to be completely precise... though now I do see they're asking for something "starting from ZF". – spaceisdarkgreen Sep 12 '23 at 04:53

1 Answers1

1

For this specific target, it's really a matter of definition and the context of metric spaces or topology or really anything other than set theory is irrelevant. When you say you have a function or a map from a set $X$ to a set $Y$, you are implicitly saying that for all $x,y \in X$, if $x=y$ then $f(x)=f(y)$. When we talk of a "correspondence" or "relation" from a set $X$ to a set $Y$, then we are not implicitly assuming anything. But then, if you don't give me the specific correspondence or relation, I won't be able to "prove" that it satisfies what you wrote, simply because it might not satisfy it.

More formally speaking, a relation from a set $X$ to a set $Y$ is a subset of $X\times Y$, and a function (or map, or morphism, etc.) $f: X \to Y$ is a relation from $X$ to $Y$, such that for all $x\in X$, there is a unique $y\in Y$ such that $(x, y) \in f$ and this $y$ is denoted by $f(x)$, and then for all $x,x'\in X$, if $x=x'$, then $f(x) = f(x')$.

Not wanting to fall into contradictions or paradoxes is a very mathematically noble goal. But specialists in set theory and/or foundations of math already provide you a lot of literature regarding what to pay attention to and what to ignore. They tell you for instance that yes the CH is independent of ZFC, that Godel's incompleteness applies, that the AC is something you might want to include in your axioms but that it's not really a necessity (for the consistency of ZF), etc. My point is, just like with science in general, there are people dedicated to the questions of foundations, and unless you are as concerned about them, you shouldn't worry much about what you're doing in relation to this topic. Just like you fly in an airplane without thinking much about how the physics of it works. Yes, it's good to be cultured enough on the foundations of math, but it's highly unlikely that there are any "dangerous" concerns that haven't been at least talked about in the literature.

I'm not discouraging you from digging deeper into things, but I'm just being realistic. Surely, many foundational questions remain open, but living with doubt is not fruitful. If studying these questions interests you more than doing casual math, then you should consider specializing there!

User
  • 409