Relativity and the Counting Process

[robert Ihnot]

I refer you to a proposal about the finiteness of the integers: http://paulandellen.com/essays/essay089.htm

How could this be? Everyone has assumed that the integers are infinite, yet a proof can be give to illustrate otherwise. Firstly we will use the characters necessary for expression of the integers in various ways: we will use the 26 letters of the alphabet along with the 10 numerals and will also throw in commas, spaces, !,^,etc, to get a collection of 50 or less characters.

Ramanujan of whom it was said, "Every positive integer was his personal friend," was asked how he arrived at a hospital, and he answered that he came by cab #65. He was told that 65 was a "very ordinary number," but Ramanujan, said that this was not true:

"65 is the smallest number that can be express as the sum of two squares in two different ways."

That, of course, illustrates that 65 has more than one name, just as 2 could be referred to as, "The smallest even integer, " or "The largest integer less than three."

Now, it has been assumed that the integers are infinite, but we will show there is an upper limit to counting, which indicates that the counting process slows down and finally reaches an upper limit, similar to that for velocity as shown by Dr. Einstein. (This is actually not all that surprising, since Relativity prevades the physical world, there is no reason not to suppose it would also effect counting, which is extrapolated from physical objects.)

Theorem: Every integer is expressible is less than 50^37 characters. Proof, by contradiction: If there is a integer that can not so be expressed, then there is a least such integer, that number is:

"The least integer not expressible in 50^53 characters."

But the number is so expressed! Thus the matter is proven and the only way to reconcile this with our process of counting is to realize that as we near 50^53, the counting processes continually slow down and eventually even stop. But this is not realizable to the counter, who is mislead by a process now classified as "number dilation." This process has frequently been confused with the fact that as the digits grow, the number takes more and more time, in general, to utter; but "number dilation" is now realized to be an entirely distinct phenomena.

 

[Hurkyl]

I have a better bound. Every positive integer can be expressed in 9 words! Assume otherwise; the smallest such counterexample would then be expressed as:

"The smallest positive integer not expressable in nine words."

Which, of course, is an expression of that integer in 9 words!


The fallacy here is similar to the liar's paradox: "This statement is false". Freedom of self-reference is bad for logic, and mathematical logic is strictly tiered so that it is not possible.

 

[M98Ranger]

The proposal presented in the link suggests that there may be a finite limit to the counting process, similar to the concept of relativity in physics. The author argues that every integer can be expressed using a limited number of characters (50 or less) and as we approach the upper limit of this character count, the counting process slows down and eventually stops. This is attributed to a phenomenon called "number dilation" which is distinct from the increase in time it takes to utter larger numbers.

While this proposal challenges the commonly held belief that the integers are infinite, it is important to note that it is based on the assumption that there is a finite limit to the number of characters used for expression. This may not be the case in all languages and writing systems, and therefore the conclusion may not be universally applicable.

Additionally, the proposal does not take into account the concept of infinity as a mathematical concept, where the idea of "counting" may not be relevant. Therefore, it is important to consider all aspects and perspectives before accepting this proposal as a definitive proof of the finiteness of the integers.