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A deductive system (as a collection of judgments and rules of inference) can be used to describe something commonly called a “set theory”. We can imagine a priori there are certain properties we would like a “set theory” - or, better yet, a “theory of mathematics” - to fulfill. For example, we hope that many of the aspects of mathematics as we know them, seem to “emerge” in such a system. This might increase the “explanatory force” of the theory - we described the same thing in less words, hence, less information.

I know there are a lot of interesting results showing how often when you try to axiomatize a certain part of mathematics, you end up implying more than you intended; and ultimately, there appear to be certain models of mathematics that recur, even under differing axiomitizations, that are hard to avoid.

I was wondering if there is an axiomitization of a “continuous interval” [0, 1] which does not require the concept of “natural numbers” as a prerequisite.

The idea would be to define “probability” as purely and minimally as possible, without building up a concept of “real numbers” and the rest of ordinary mathematics first.

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  • Would the Geometry Axioms be what you are looking for? Geometry has points, and lines, and line segments. Commented Apr 23 at 14:07
  • In ZFC, one can construct the empty set, the Successor Function, and Dedekind Cuts, and then say afterwords that these sets are a constructions of numbers. But, the Axioms don't make any mention of Natural Numbers...beyond the fact that we enumerate our list of variables with natural numbers. But we dont have to do that, its just for convenience. I'm not sure I entirely understand your question.How exactly do you envision a potential solution to construction the interval [0,1] without natural numbers, where 0,1 themselves are natural numbers? Do you mean the interval (0,1)? Commented Apr 23 at 14:12
  • i.e. can we construct a notion of the reals, for reals which are not natural numbers- without the notion of natural number??? I think that would be, a interesting question, I think one approach, is to ask if we can construct the notion of integer, and rationals first. Commented Apr 23 at 14:13
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    There is Tarski's axiomatization of the reals that bypasses natural numbers. It is decidable which means that full arithmetic is not expressible in it. Concerning probability, Mumford proposed to put random variables directly into foundations "resulting in a more intuitive and powerful formalism" because "it is artificial and unnatural to define it in terms of measure theory."
    – Conifold
    Commented Apr 23 at 17:52
  • I learned plane geometry of a number line in grade school - I think grade 5. The teacher draws a line segment on the blackboard. By definition (assertion) the line segment is composed of an infinite number of points. By definition the line has infinite in length. That means each segment in the line, no matter how large or how small, is composed of an infinite number of points. By definition or axiom the points are infinitely small. None of this involves the specification of numbers. To create a coordinate system on the interval 0, 1 we put "0" and "1" on any arbitrary segment length of a line. Commented Apr 23 at 19:47

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Is there a set theory which implies the interval [0, 1] but no more?

it's weird to talk about 'implying an object', as implication is a connective, something that holds between propositions, not objects. could it be 'implies the existence of...'? in any case, one would also need to know what 'more' is supposed to mean here

I was wondering if there is an axiomitization of a “continuous interval” [0, 1] which does not require the concept of “natural numbers” as a prerequisite.

for some purposes, yes, as for example here

The idea would be to define “probability” as purely and minimally as possible, without building up a concept of “real numbers” and the rest of ordinary mathematics first.

as for synthetic/categorical probability, the main line of development seems to be lawvere's 1962 "category of probabilistic mappings", bogdan's 1976 "new approach to probability via algebraic categories", giry's 1982 "categorical approach to probability theory", fric & papco's 2010 "categorical approach to probability", adachi & ryu's 2019 "category of probability spaces", and fritz's 2020 "synthetic approach to markov kernels, conditional independence and theorems of sufficient statistics", but naturally some of these assume the ordinary reals (plus measur[abl]e spaces, topology, and whatnot) from the go

but perhaps most importantly: 'finitary probability' is (a subset of) finitary combinatorics, which is (the same as) finitary arithmetic, which IS a set theory, so one shouldn't be surprised that for this long people have regarded infinitary/higher-order probability as belonging/being (some subset of) infinitary/higher-order arithmetic, aka analysis and set theory

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    I think most categorical/synthetic approaches will assume the reals exist. In particular, giry, adachi, and most work in synthetic measure as of 2020 will build upon the reals. Esp since they are not motivated by using only the interval. Actually, I don't know of a single approach that doesn't assume the reals, but I am not familiar with fritz or fric
    – emesupap
    Commented Apr 23 at 16:40

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