zylo

f(n) =14159.. to n places is defined and a natural number for all n.

So it is defined for infinite n with infinity defined as non-finite (it is not defined only up to, say 10)

v8archie

So you now think there is an infinite natural number despite it going against the definitions?

When we define something for all natural numbers $n$, it means that there is no finite upper limit on $n$. It can get arbitrarily large, but remains finite.

skipjack

Consider this (non-terminating) sequence of natural numbers:
1
14
141
1415
14159
141592
1415926
.
.
.
On each successive line of the above, one further digit of the decimal expansion of $\pi$-3 (without the leading ".") is given. Similarly, any number between 0 and 1 corresponds to a sequence of natural numbers. This does not tell you whether or not the set of all numbers between 0 and 1 is countable.

v8archie

There are two critical points to skipjack's post:

1. $\pi$-3 is not an element of the sequence he presents; and
2. there are uncountably many such sequences.

skipjack

Although that second point is true, I didn't include a proof that it's true.

That's correct, but the fact that f(n) is a natural number for every natural number n does not imply that the non-terminating sequence 1415926... is a natural number. You've already defined a natural number here, where you stated explicitly that it has a number of digits, n, that must be finite (which isn't true of a non-terminating sequence).

zylo

1,2,3,4,5,......... is a non terminating sequence.
.33333........ is a non-terminating sequence:
.3(1).3(2).3(3).3(4).3(5)

This is what I said in your reference:
"The definitions hold for every finite n, but not for n=∞ (undefined)."

Unless you take the definition of infinity as "for all n," as illustrated below:

If ri is a decimal digit,
r1r2.....rn is a natural number for all any n.
.r1r2.....rn is a decimal fraction for all any n.

1:1 correspondence between decimal fractions and natural numbers:
0$\displaystyle \leq$x<.1 add .1 and remove decimal point:
.00001...->.10001->10001
.1$\displaystyle \leq$x<1 remove decimal point:
.14159....->14159.....
.333333.....->3333....

Remove trailing 0's.

skipjack

You chose to provide a definition of any natural number, n, that was finite. You then stated that your definition was valid for every finite n (as being finite and having a finite number of significant digits amount to the same thing). You also stated "but not for n=∞ (undefined)" . . . what exactly did you mean by that?

Do you agree that your statements appeared to agree that every natural number is finite?

Do you agree that you haven't provided any definition of a natural number that isn't finite?

Do you agree that all the accepted definitions of a natural number that were in use when Cantor was alive require that such a number is finite?

What do you mean by "all any"? Where has anybody else used it?

Do you agree that the index variable used to reference any individual member of an enumeration (as that concept was understood in Cantor's time) was required to have a finite value?

v8archie

That's not a definition that any mathematician recognises. I don't think it even makes any sense.

An infinite thing is the complement to the set of all finite things in the set of things. So an infinite sequence is a sequence that is not finite.

A finite sequence is characterised by having a finite number of elements. It terminates. An infinite sequence, being the complement of this set, therefore does not terminate.

zylo

Finite: A natural number

Infinite: All the natural numbers

"The natural numbers are finite" is ambiguous, and hence tricky.

If it means individually, it is correct. If it means collectively, it is incorrect.

v8archie

What you have written (in that post) is mostly correct but they are not definitions.
It's not ambiguous. The natural numbers are finite (that's a property that every one of them holds). The set of natural numbers is infinite (that's a property of the set, not of its elements, which are natural numbers).