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I asked recently what a differential equation is, formally. However, I now realize that I should have asked a more basic question first. My question is, what is the formal, rigorous definition of an equation? I want a definition in terms of set theory. That is, it should be a definition that can, at least in principle, be unwound to a first-order formula of ZFC set theory.

Henry
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user107952
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    First you are going to have to decide whether you are talking about: something like "$\forall (x,y) \in \mathbb R^2: (x+y)^2=x^2+2xy+y^2$" or something like "$x+1 = 3$ has the solution $x=2$"? – Henry Dec 10 '21 at 11:39
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    There's multiple different definitions: ${ x \in X | f(x)=g(x) }$ or ${ (x,y)\in X\times Y | f(x)=g(y) }$. The first one is called equalizer, and the 2nd one is a pullback. – tp1 Dec 10 '21 at 12:25
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    An equation is a symbolic expression in the language of some mathematical discipline: arithmetic, algebra, calculus. Usually, to "solve" an equation $s(x)=0$ means to find if there are some values for $x$ such that the equation is satisfied. – Mauro ALLEGRANZA Dec 10 '21 at 12:27
  • In model theory and universal algebra, an equation in a functional signature $\Sigma$ is a sentence of the form $t=s$ for terms $t,s$ in that signature. (Note that there is some conflation between an equation and the universal closure of an equation, e.g. "$x+y=y+x$" versus "$\forall x,y(x+y=y+x)$.") See also terms like equational logic (which is really the logic of universal closures of equations, but meh). – Noah Schweber Dec 10 '21 at 16:24
  • @NoahSchweber So, that raises the question, how does one define what a sentence is, formally, in model theory? – user107952 Dec 10 '21 at 20:17
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    @user107952 A sentence is a finite sequence (= map with domain $\in\omega$) of symbols (there are various ways to specify a symbol set for FOL) satisfying certain combinatorial properties (e.g. "parentheses match up"). This isn't hard to whip up at all, but it is rather tedious; it will basically look a lot like the setup of Godelization. – Noah Schweber Dec 10 '21 at 20:54
  • @NoahSchweber Perhaps you should post your comment as an answer. That is exactly the kind of thing I am looking for. – user107952 Dec 10 '21 at 20:59

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Per the comments, let me say a bit about how one can - in $\mathsf{ZFC}$, for example - talk about (say) first-order sentences in a rigorous way. I'm not going to fill in all the details because ... whoo boy ... but I'll hopefully make the situation clear.

Our first step is to define the term non-logical symbol. Here's where the ugly happens:

  • "$x$ is an $n$-ary function symbol" will be shorthand for "$x=\langle 0, n, a\rangle$ for some set $a$." (I'm thinking of constant symbols as nullary function symbols.)

  • "$x$ is an $n$-ary relation symbol" will be shorthand for "$x=\langle 1,n,a\rangle$ for some set $a$."

  • "$x$ is a non-logical symbol" will be shorthand for "for some $n\in\omega$, either $x$ is an $n$-ary function symbol or $x$ is an $n$-ary relation symbol."

We then brute-force carve out some logical symbols:

  • "$x$ is the $=$ symbol" will be shorthand for "$x=\langle 2,0\rangle$."

  • Etc. (the idea should be clear).

Finally we have the variables:

  • Our variables will be sets of the form $\langle 3,a\rangle$ for some set $a$.

Terms, formulas, sentences, and proofs can now be represented by sets in a well-defined, if incredibly tedious, way. E.g. "Every formula has as many $($s as $)$s" can be made precise via the above approach and proved in $\mathsf{ZFC}$ or indeed much less. Similarly we can directly talk about structures: if $\Sigma$ is a set of non-logical symbols, a $\Sigma$-structure is a pair $(A,b)$ where $A$ is a (usually nonempty) set and $b$ is a function with domain $\Sigma$ satisfying [usual list of properties]. Defining $\models$ is a bit harder, but see this old answer of mine. Note that here we actually do have to "do some math," and (satisfyingly (pun intended)) what saves the day is the perhaps-at-first-glance-circular-and-certainly-crtiticizable-on-philosophical-grounds-if-nothing-else Tarskian definition of truth.

Putting all this together, we have a perfectly straightforward - if, again, incredibly annoying - way to implement our usual model-theoretic arguments in $\mathsf{ZFC}$.

Noah Schweber
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  • Can't we just work in the metatheory and treat them as propositions? – Poscat Oct 18 '23 at 04:32
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    You did what the OP asked, and the OP accepted this as the answer, but my impertinent self wonders just how useful he/she actually found it. I rather suspect what such an inquiring student might find more useful is a clear specification of the rules of inference that deal with equality in formal deductions (most students will never be exposed to this! and maybe most mathematicians couldn't tell you!). – user43208 Feb 17 '24 at 22:54
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The term "equation" doesn't need a set-theoretic definition, because the "=" symbol is already part of the underlying first-order logic. We informally call a sentence an equation if it has the form $A=B$ or $\forall x (A(x)=B(x))$ or any of a variety of other forms that in spirit "state an equality". The axiom of extensionality specifies a relationship between $=$ and $\in$.

Karl
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