Actual Infinity in Reality?

[polymath257]

This is just false. Mathematicians are talking about actual infinities. They even talk about different sizes of actual infinities (for example the difference between countably and uncountably infinite sets).

There is no requirement of having an 'open end' in the description of a set:

N={x: x is a natural number}

is a perfectly well defined, infinite set. No 'open side' and no 'bounded side'. The description in terms of a list,

N={1,2,3,4,...}

is more a convenience for those who cannot read mathematics than anything else.

What you seem to think is that the second is some sort of process. It isn't. It *does* appeal to your understanding to know what things are in the set and what are not. But the list itself is just a notational convention and nothing else. In the same way, we can define an uncountbaly infinite set

R={x: x is a real number}

or

[0,1]={x: x is a real number and 0<=x<=1}.

Both of these are uncountably infinite sets.

There is no such thing as a 'potentially infinite set' in math. Sets are either finite or infinite. In the latter case, they are actually infinite. They can be countably infinite or uncountably infinite. For the uncountbaly infinite sets, there are infinitely many different possible cardinalities (although very few are used in practice)

Look in *any* math book and you will find NO distinction made between potential and actual infinity. The reason? Those notions have been replaced. They are no longer used because they are confused and ill defined.

So, I will make a challenge. Look in *any* advanced level math book produced in the last half century. Find *any* reference that discusses *at all* the notion of 'potential infinity'. I challenge you to find a single source *in math* for your claims from the last 50 years (I'll even go 75 years).

Okay, I was wrong about what mathematicians consider their sets to be. They do consider them to be actual infinite. But what does that mean? It does not mean the same thing a physicist (or even a philosopher) would mean with the same phrase. 

However, my point was better said in my post from page 6:

In the philosophy of mathematics, the abstraction [defined below] of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity {emphasis added}

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

In fact, regarding the Axiom of Infinity article, later on the article says: 

Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation...

I am not claiming the merits of one set of axioms or another--only illustrating that axioms are assumptions and are not self-evident truths (more below on this). 

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics) {emphasis added}

So, putting braces {} around 1, 2, 3, 4, 5... makes a new mathematical object: A set. This is not a thing found in the real world that they needed to describe. This is an abstract object  created by axiom. More specifically, a non-logical axiom: 

As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question...in the philosophy of mathematics.[5] https://en.wikipedia.org/wiki/Axiom#Mathematical_logic {emphasis added}

Further, when mathematicians say "exist", they do not mean the same thing as nonmathematicians:

...The use of the word "exist" is merely a grammatical convenience here; mathematicians and nonmathematicians do not mean quite the same thing by this word. Unfortunately, we mathematicians don't have a better word; to be more precise we would have to replace this one word with entire paragraphs. If we assume the Axiom of Choice, we are not really stating that we believe in the physical "existence" of those sets or functions. Rather, we are stating that (at least for the moment) we will agree to the convention that we are permitted write proofs in a style as though those sets or functions exist.

Whether those sets or functions "really" exist is actually not important, so long as they do not give rise to contradictions. Mathematicians are perfectly willing to use devices that may be fictional, as intermediate steps in getting from a real problem to a real solution. https://math.vanderbilt.edu/schectex/cou...ntial.html

This last one is actually an easy to read article that explains things in plain language. 

THEREFORE, because infinite sets are a results of non-logical axioms (assumptions) that are not self-evidently true AND, the term 'exist' is not the same in mathematics, I stand by my claim that infinite sets in mathematics are not an indication that an actual infinite series or objects can exists in reality.

But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.