Infinite Digit Numbers
is there more than one?

O   
n another page, I indicated that there were an infinite number of non-Cantorian type numbers with neither beginning nor end, i.e. qualitative numbers denoting the order of symbols such as this.

...857394883623950684637...
This may have been incorrect. I believe it is more likely that there is only one such number.

Non-mathematical symbols

The only real abstract representation of a quantity (a number) would be a series of identical dots which correspond to any specific integer (quanta) ... like this ...

....... = 7
Here, the "7" is the unacceptable symbol and the dots are the "real" abstract number. "7" is a convenience as it would be horrendous (as a practical measure) to use hundreds of dots to denote numbers. Now, if we still wish to accept the "7" type number symbol as logically viable ... we may have a set of numbers that is greater than aleph0 (by Cantor's diagonal proof). Similarly, to show that there are an infinite number of "qualitative numbers" with no beginning or end as above, we could say that there are any number of symbols that denote integers rather than just dots. We could have not only base ten (with ten symbols) but any base whatever. Then we could infer that there is more than one such number by defining one such number as one without the symbol "7". And hence, any other number with a "7" symbol in it would necessarily be different. Or, we could define an infinite digit number as one with only two "7"s in it and so on.

This is a cheat however.

If we denote such numbers with the minimum number of symbols needed to convey randomness, we end up with only base two, i.e. ones and zeros (...000101011100101010101...). Now we are left to define, in terms of O's and 1's a number which is different from all other numbers containing an infinity of digits.

How could we possibly do this?

Because we are purporting a random and infinite sequence, we can only say that two such numbers are different at some finite length within the infinite sequence, e.g. ...001011111000110... is different from ...101001010100101... But how can such a distinction be proved in any way by a finite mechanism? Certainly, the above sequences must occur somewhere in each infinite number. Hence, we cannot, in principle "prove" that one is different from another.

By the principle of logical economy, they are therefore the same number. There is only one such qualitative number with an infinite sequence of digits with neither beginning nor end.

Of course, we might add that such a number with only an infinite string of 1s (...111111111...) would qualify as another distinct number with infinite "digits". But such a number can only be "defined" as different from a random sequence ... we cannot generate such a "different" number by any feat of mathematical induction. For, by the same reasoning as above, we could only examine a finite string of 1's ... and ... that finite string, no matter how large, is certain to occur in our random string with an infinite number of 1's and 0's.

Again, there can then be only one such number. As long as an infinite string of symbols exists relative to a finite string, it is incumbent on us to accept it as THE only one of its kind. And if the universe is the embodiment of logic as I suppose ... then ... the integers (the finite quantities of dots) are embedded in that one infinite qualitative number. Hence, space ... the geometric embodiment of such a number ... is continuous, non-discreet, non-divisible ... just as we would expect to find it.

See,
EBTX - The Structure of Space



Ebtx Home Page