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Defining Bayes’s theorem from scratch in ZFC

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I have tons of interrelated questions which I would like resolved in order to help me answer this Philosophy SE question about Bayes’s theorem given a certain probability of 0.

I’ll assume I am working in ZFC, which I have tons of questions about. In fact, I am still seeking even a basic understanding of how ZFC works.

ZFC can be formulated as a collection of 9 formation rules written in the language of first order logic. I read somewhere that somehow, FOL implies the existence of at least one “thing”, so you do not need to assume the existence of the empty set as an axiom, for ZFC.

The important thing about first order logic is that it has quantifiers.

The formation rules of ZFC can be written using FOL, as (roughly):

For all z (in some universe), if z is in x, and z is in y, then “x = y”. (I am pretty sure that, since FOL includes “relations” as a syntactic feature, “set membership” is actually nothing more than some arbitrary “relation” (and so is “equality”, and so is the implication arrow.) Thus, what we have here is an interaction between three relations?: for all z in some universe, if xR1z R2 yR1z, then xR3y. And this is true for all x and y in this universe, so we also use the universal quantifier over them.)

I’m going to skip ahead since there’s so much to write and learn. The other axioms include well-foundedness (roughly, sets are disjoint with their elements), the axiom (“schema”) of restricted comprehension (that for any formula of FOL, the set of all elements of some set S meeting that formula exists), the axiom of pairing (or, “the ability to put things together”), the capability to take unions and power sets, the existence of any sets under any definable function, and actually I would like to reject the existence of an infinite set if possible, and I read that therefore I do not need the axiom of choice.

A more intuitive way for me to summarize this is basically, “preconditions”: equality and regularity; and ways to generate new sets: power sets, unions, pairings, and “logical conditions” (comprehension). (I would like to know why pairing could not be accommodated by unions, and why the function axiom (“replacement”) is needed, since I thought to specify some function, it would require the existence of the set to begin with.)

Basically, my wish was then to define “probability” and “conditional probability” and then prove Bayes’s theorem. Perhaps someone can assist me with that. Thank you.

I know this question is full of different questions so I will probably break it into individual sub-questions as new posts.

set-theory zfc bayes-theorem

posted 3 months ago

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Julius H.‭

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Derek Elkins‭ wrote 3 months ago

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FOL is a specification language and quantifiers and logical connectives, including implication, are what you use to write the specification. You introduce predicate symbols, such as $\in$, to give a name for the things you are describing. Equality is usually taken as part of FOL, but you could also treat it as another predicate you're specifying. For most purposes, it doesn't make a difference which you choose. In typical, minimal presentations of ZFC, then, the only predicate symbol is $\in$. (I usually reserve "relation" for the set-theoretic concept, i.e. a subset of $X\times Y$ for given sets $X$ and $Y$.)

Traditionally, the domain for a FOL theory is assumed non-empty. This can either be via an explicit axiom, i.e. $\exists x.\top$, or, often, it is consequence of a sloppy formulation of the quantifiers. Either way, this is mostly done for meta-logical convenience. That said, the Axiom of Infinity does assert the existence of a set, so this isn't an issue for ZFC.

Julius H.‭ wrote 3 months ago · edited 3 months ago

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Thanks a lot. I’d love to go through what you just said and perhaps have an extended conversation about this. That way, I could hopefully go back and revise and improve my post.

“FOL is a specification language.” Could you elaborate on the significance of this? In other words - a specification language, as opposed to something else?

You say set membership is a predicate rather than a relation? Is that because relations (and functions) are technically defined as predicates, in FOL? For example, to say that xRy is actually to say that R(x, y) is True?

You say “equality is usually taken as a part of FOL”. Is equality taken as a primitive notion, then, as opposed to defined as a relation? (I suppose I can ask that question on its own, in another post.)

I usually reserve "relation" for the set-theoretic concept, i.e. a subset of $X \times Y$…

I suppose I thought relations were a primitive notion in FOL since I thought I saw that presentation once in a text on model theory.

Julius H.‭ wrote 3 months ago

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the Axiom of Infinity does assert the existence of a set, so this isn't an issue for ZFC.

I see. Perhaps I misremembered what I had read somewhere - that it is not FOL that implies the existence of “something”, but rather, the axiom of infinity. That’s very helpful. Thank you.

The axiom of infinity is quite interesting. I did not realize before that there is basically an axiom that generates all the ordinals. Isn’t that basically what it does?

Another question since we’re here. I did not understand why the axiom schema of restricted comprehension requires an axiom “for each $$\phi$$”. Does it say that for every possible sentence $$\phi$$ of first-order logic, and for any set $$S$$, there exists a set whose elements are the elements of $$S$$ for which $$\phi$$ is true - but why does that not count as a single axiom? Is it because you’re quantifying over the sentences of FOL, which technically isn’t something FOL can express about itself (ie the variables of FOL only refer to sets?)

Derek Elkins‭ wrote 3 months ago

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I used "specification language" to emphasize that FOL is a complete system by itself and to emphasize the distinction between the language in which we are specifying something, i.e. FOL, and the thing being specified, in this case ZFC, or, more precisely, the $\in$ predicate. I call it a "specification language" because you can't construct things in FOL, only describe them. At best you can assert that something should exist. In contrast, type theory and the "language" of set theory allow you to construct things.

"Relation" is an ambiguous word that is often used for "binary predicate" but is also often used for "subset of a cartesian product". "Predicate" is usually only used for the logical concept. It's useful and important to keep these notions distinct.

I (and others) discuss the nature of equality in FOL here on Math.SE.

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Comments on Defining Bayes’s theorem from scratch in ZFC

Defining Bayes’s theorem from scratch in ZFC

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