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In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing the corresponding (restrictive) conditions on it.

In my opinion this is not satisfactory since an arbitrary set-theoretic function $\mathbb{R}\to\mathbb{R}$ is a “horrendous”, too wide and unstructured object (that does not mean that an arbitrary continuous function $\mathbb{R}\to\mathbb{R}$ is not “horrendous”, but there is less of them for starters).

My question.Is there some way to reframe the foundations so that there is a "intrinsic" way to define continuous/smooth/analytic functions $\mathbb{R}\to\mathbb{R}$? I want such a definition where a continuous/smooth/analytic function is not a set-theoretic function having a particular property but an atomic notion.

This is in some ways similar to the following question which asks if a particular homotopy type can be defined in HoTT without replicating the set-theoretic definitions.

  • Could you be a bit more specific on what precisely you find unsatisfactory? Starting with an extremely general structure and then imposing additional requirements and stuctures in order to obtain a notion that fits our purposes appears to me to be the basis of success of 20th and 21st century mathematics. A few examples: set -> power set -> topological space -> compact Hausdorff space; or: group -> topological group -> Lie group; or: vector space -> normed vector space -> Banach space -> Banach space with Schauder basis. I'm not sure that I find anything unsatisfactory about this approach. – Jochen Glueck Aug 13 '20 at 13:58
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    The problem here is that any definition of continuous functions would be of the form "a function $f$ is continuous iff...", and that presupposes talking about functions $f$ in more generality beforehand. –  Aug 13 '20 at 14:09
  • Also, since you tagged model-theory and you mentioned foundations: are you familiar with smooth infinitesimal analysis / synthetic differential geometry? There infinitesimals are postulated to exist axiomatically, and logic is modified to remove excluded middle, and then axiomatically all functions are continuous and differentiable. – Willie Wong Aug 13 '20 at 14:09
  • If you want to solve these issues with new axioms instead of new definitions, then @WillieWong's suggestion is the way to go: the axioms of smooth infinitesimal analysis for a world where functions are all smooth, or the axioms of Bishop-style constructive analysis where it is consistent that all functions are continuous. –  Aug 13 '20 at 14:22

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The question is vague enough to make it less than clear what kind of reply you would accept as not describing a horrendous object but hopefully polynomials would fit the bill. One can then define your required spaces (also Lebesgue) spaces as the completions under suitable metrics or uniformities. Of course, one then has to show that the resulting spaces consists of functions (or equivalence classes thereof in the Lebesgue case) but this is a fairly simple and illuminating exercise. Just where this rates from $1$ to $10$ on your scale of horror is something only you can know.

user131781
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This is sort of an extended comment, to find out what you mean by intrinsic.

For the special case of analytic functions, you can use power series.

Definition An element of $C^\omega(\mathbb{R})$ is a countable set $f$ of elements of the form $(x, r, (c_i)_{i\in \mathbb{N}})$ such that

  1. The union of the open intervals $(x-r, x+r)$ over the elements of $f$ cover $\mathbb{R}$.
  2. For each element of $f$, $\limsup \sqrt[n]{|c_n|} < 1/r$. (And hence the power series $\sum c_n (y-x)^n$ converges on $(x-r, x+r)$.
  3. If $(x_1 - r_1, x_1+r_1) \cap (x_2 - r_2, x_2 + r_2)$ is non-empty, then for any $y$ in the intersection $\sum c_{1,n} (y - x_1)^n = \sum c_{2,n} (y-x_2)^n$.

This definition is practically the same as the usual definition of analyticity of holormorphic functions, except that where usually one specifies "$f$ is a function such that its power series has non-zero radius of convergence at every point" we replace by "$f$ is an object with power series with non-zero radius of convergence at every point, such that on over lapping intervals the power series based at different points agree [making it a function]."

It is possible that while this definition seems to meet the letter of your question (by not defining $C^\omega(\mathbb{R})$ as a subset of $\mathbb{R}^\mathbb{R}$), it may be against the spirit of your question. In that case, please edit your question to provide more info on what is allowed by "intrinsic".

Willie Wong
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