Defining Bayes’s theorem from scratch in ZFC
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.
1 answer
All of this has already been dealt with on Metamath, see mpeuni/bayesth or asrt/bayesth.
At first, they had to decide on concrete formulations of ZFC axioms, which must be stated in a formal language. All referenced axioms (including definitions of, for example, the class of probability measures and conditional probability) are linked at the bottom of the site of the first link, and you can explore the whole proof by following the references to subproofs.
Note that these proofs are fully formalized and thus can be verified by a machine. This is always the case when deriving everything from axioms, rather than making intuitive statements in natural language, which are rather social proofs in contrast to formal proofs, as noted here.
It appears to me that you underestimate the complexity of creating formal proofs from scratch, which is indicated by the mere amount of definitions that this theorem relies upon. Even though the theorem contains only one name of the person who completed it, it resulted from a big collaborative effort due being built upon this large framework.
Formal proofs work purely on syntax (i.e. structure) and not at all on semantics (i.e. meaning). Outsourcing the thinking process (which is susceptible to errors) to formal methods is precisely what axioms and proofs are meant for. Social proofs (which are most common in mathematics) just do not go all the way.
Which is mainly, because that would be too hard for most mathematicians to both create and conceive. After all, mathematics de facto is a social framework for humans to intuitively understand abstract objects.
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