# Does a Cycle Based Alternative to Set Theory Exist?

My (limited) understanding of Mathematics in general and of Set Theory (being a widely accepted foundational system of Mathematics) in particular, is that the mathematical objects described therein exist in their entirety at any given instant. It is also my understanding of Set Theory that the entire thing is built on the idea of objects, real or abstract, that can be included in or excluded from these sets based on the characteristics of these constituent elements.

At this point, it occurs to me that there is no such thing (in the physical world) as an object. Many will disagree with me on this point, but I'm fairly certain that current quantum theory has pretty conclusively shown that even the most fundamental of particles still have a wave nature. That is to say, they are cyclical. In the physical world at least, any "set" created out of physical objects must also have a duration (or a wavelength?) because all of the included elements must also have this characteristic.

Does there exist an alternative formulation of, or alternative to, traditional axiomatic set theory that deals with collections of "cycles" instead of collections of "objects"?

********** EDIT *************

I'm not sure if I'm supposed to edit the original Title of my post/question or not, so I'm restating my original question below.

Original Question: Does a Cycle Based Alternative to Set Theory Exist?

Amended Question: Does a foundational mathematical theory exist which is based on collections of processes as opposed to sets of abstract mathematical objects?

Perhaps I shouldn't have tried to include my set of quantum "particles" to try to illustrate my question... I think it just confused the point I was trying to get at and for that I sincerely apologize.

Also, I should have used the term process instead of either cycle or wave. As many have pointed out in the comments below, the sets of set theory most often involve sets of abstract mathematical objects, often with no regard for the order of the items within those sets. But how does one know what is contained within a set?

Even if a set is an abstract mathematical object, how do you know for sure that somebody hasn't added another abstract mathematical object to that set since the last time you checked (I know that when I'm writing a function where the size of an array or list is important, I always count it. I never assume that I know how many elements are in that array/list/set)?

How can you count the number of elements contained within a set without the act of counting those elements? Is the act of counting not a process carried out over time? And what is arithmetic if it's not counting? And if arithmetic is at the very core of math, and counting is at the very core of arithmetic, and counting is a process that takes time (cycles) to perform, then how can/does set theory justify the idea that an entire set can exist, represented as some kind of etherial eternal platonic form, without the act of instantiating that set, even if it's just a symbolic instantiation?!

And I know if seems like I'm totally talking my self into saying that computable set theory is the answer to my question, but I'm not entirely sure that it is... Did I clarify anything with this edit? I hope so, but I'm not entirely convinced.

• Mathematical theory of sets does not developed from some (maybe impossible) definition of object and collections of objects, but from the wide use made by mathematical theories of the concept "collection of...": number, functions, lines, etc. Mar 3, 2020 at 7:43
• In the same way, IMO, we have to start not from a (maybe impossible) definition of "cycle" but from the identifications of some relevant "contexts" where mathematical theories use the concept "collection of cycles"... if any. Mar 3, 2020 at 7:45
• "Objects", as understood in mathematics, would include what you call "cycles", in particular, waves (or any distributions that do not have to be wavy at all) are "objects". If one wants to include "duration" there is no need to represent it at the fundamental level, one can simply consider pairs, of a set (or whatever) and a number ("duration"). But according to quantum field theory, which supersedes quantum mechanics and is relativistic, "duration" is not Lorentz-invariant and hence has no physical significance. Mar 3, 2020 at 12:42
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– J D
Mar 3, 2020 at 13:59
• Sounds like you asking about categories and topoi. Give this WP Foundations o mathematics article a read.
– J D
Mar 3, 2020 at 14:01

I would like to paraphrase what you wrote:

• My limited understanding of set theory is that the mathematical objects described in sets exist in their entirety at any given instant in time.
• Quantum theory has pretty conclusively shown that even the most fundamental of particles still have a wave nature.
• things on a wave nature don't exist at just one instant in time.
• Is there an alternative to set theory where everything in the set is a wave instead of an "object". For example, if `x` is in a set, then `x` is sine-wavey thing.

I think you would understand sets better if you saw some examples.

Let `S` be the set `{1, 2, 99}`

• set `S` contains the number 1.
• set `S` contains the number 2.
• set `S` contains the number 99.
• set `S` does NOT contain the number 4
• set `S` does NOT contain the number 50
• set `S` does NOT contain the number 72349
• set `S` does NOT contain the number 50

In physics, photons are are model used to describe light... (sunlight hitting your face, laser beams, etc...). Photons are modeled as having a "wave-particle duality."

The number 7 is not a photon.

Look... suppose you play chess. Chess is a game having rules.

You can play chess:

• Using a wooden chess board, stone chess pieces, and move the pieces with your fingers (no computer)
• You can play chess over the internet, using a computer, where there is no wood, and no stone.

The rules of chess are the same either way. The rules of chess are NOT a collection of photons having "wave-particle duality."

You can encode the rules of chess in many different mediums.
You could write the rules of chess down on a clay tablet, in German, and put the tablet in a really dark room (no light implies no photons)
You could write the rules of chess down, and post them on a website.

I am paraphrasing here, but you said something like, "the objects described in sets exist in their entirety at any given instant in time. "

It is very VERY incorrect math to say the following:

• On January 10th, 2019 set `S` was {1, 2, 99}
• We added the number 69 to set `S` the next day on January 11, 2019
• Set `S` currently contains the number 54
• Set `S` used to NOT contain 54, but now it does contain 69.
• Set `S` is currently `{1, 2, 54, 99}`

Sets have nothing to do with time.
It makes no sense to say that the objects in a set "exist in their entirety at any given instant in time."

Computer scientist sometimes talk about time.
Suppose we are sorting a list of several letters. We want to put the letters into alphabetical order

`````` VALUE [H,  C,  E,  A,  I,  B,  D,  F,  G]
INDEX  1   2   3   4   5   6   7   8   9
``````

A computer scientist might want to talk about the set of indices which have been sorted at any point in time.

The solution: give each set a different name.

• On step 1 of the algorithm the set of sorted indices is S1 = {1}
• On step 2 of the algorithm the set of sorted indices is S2 = {1, 2}
• On step 3 of the algorithm the set of sorted indices is S3 = {1, 2, 3}
• On step 4 of the algorithm the set of sorted indices is S4 = {1, 2, 3, 4}
• When we are done, the set of sorted indices is {1, 2, 3, 4, 5, 6, 7, 8}

Note: the sets are not all named "`S`" Each set gets its own, separate, name.

One of the key features about sets is that order does NOT matter.

The following two sets are the SAME:

``````{H,  C,  E,  A,  I,  B,  D,  F,  G}
{A,  B,  C,  D,  E,  F,  G,  H,  I}
``````

"A4Q-93Z" is NOT the same license plate number as "349-AQZ."
However, the following two sets of characters are the same set.

``````{"A", "4", "Q", "-", "9", "3", "Z"}
{"3", "4", "9", "-", "A", "Q", "Z"}
``````

The following sets are all equivalent:

``````Bob = {1, 2, 3}
Sarah = {1, 3, 2}
Ian = {3, 1, 2}
U = {3, 2, 1}
X = {2, 3, 1}
apple = {2, 1, 3}
``````

For example, `Bob = Sarah` and `U = Ian`

Why on earth would you want each element of a set to be a "wave"? That's just weird.

I guess you can do it:

Let `f` be a function such that:

• The possible inputs to `f` are any real number.
• For example, `4` is a valid input to `f`.
• `949.23223` is also a valid input to `f`.
• For any real number `x`, the output of `f` is `f(x) = 9*sin(x) + 354`

Okay... let Gozer the Destructor be the set `{sin, f}`.

It follows that Gozer the Destructor is a set of mathematical waves.

• I truly appreciate your enthusiasm, but it seems like maybe I wasn’t clear about what I was really trying to get at… I'm going to edit my original post to be more clear and to try to address some of the points you raised. And, after reading your objections, I think I'm better able (think you btw) to articulate the main thrust of my original question. Something more like: Are there any theories that treat the sets as processes rather than as abstract mathematical objects? Mar 5, 2020 at 5:23

“Cycles” is quite specific, but you might find Category Theory interesting!

https://plato.stanford.edu/entries/category-theory/

There is a large branch of mathematics whose practical aim is (broadly speaking) the figuring of maths and proofs in more algebraic, combinatorial terms, and it tries to use Categories (whose ‘elements’ are directed arrows, like in Graph theory) as a key framework for discussions about mathematical structures.

• That's heading in the right direction, but not what I was trying to get at... Category theory and computable set theory are both closer than traditional set theory to what I'm asking about. I'm going to edit my original post and try to better articulate what I was trying to ask. Mar 5, 2020 at 5:26

Well, I think it's time to close this one down.

The closest thing I've found to what I was originally asking about is a sort of mashup of Operationalism and Aristotelian Realism. Apparently there's also a Euclidean Arithmetic that explicitly attempts to frame mathematics in terms of "real" objects/operations grounded in physical reality.

The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". ~Wikipedia

I guess I just always assumed that all Mathematicians fell into the Mathematical Platonism school of thought. It turns out that they do not. Well, you know what they say about making assumptions...