Paradoxical Proof?

[Bridge]

Actually there has to be a smallest natural number N (which we of course don't know).

I do not see how it is even remotely possible to describe something indefinable.

Wait, what? All I meant was that there has to be a smallest natural number N which can be described with fourteen words or fewer.

There very well may be, but at the present moment it is impossible to know this "smallest number" - therefore it strikes me as completely impossible to consider it. About as impossible as mapping the coordinates of the edge of the universe.

[BAndrew]

Actually there has to be a smallest natural number N (which we of course don't know).

I do not see how it is even remotely possible to describe something indefinable.

Wait, what? All I meant was that there has to be a smallest natural number N which can be described with fourteen words or fewer.

There very well may be, but at the present moment it is impossible to know this "smallest number" - therefore it strikes me as completely impossible to consider it. About as impossible as mapping the coordinates of the edge of the universe.

It's indeed impossible to know this number, but we can easily prove that such a number exists.

Spoiler below!

Assume S the set of all natural numbers that can't be described with 14 words or less (which is not an empty set and is a subset of N). This set must have a minimum natural number N because:

Let another number X of the same set.

If X<N then there is a smallest number (it's just not N)
If X>=N then N is the smallest number

But I think I missed the point you are making.

[Bridge]

Spoiler below!

Assume S the set of all natural numbers that can't be described with 14 words or less (which is not an empty set). This set must have a minimum natural number N because:

Let another number X of the same set.

If X<N then there is a smallest number (it's just not N)
If X>=N then N is the smallest number

This is a valid proof? Because it seems to me like yet another attempt to apply the concept of infinity to a system designed only to work with finite numbers.

[BAndrew]

Spoiler below!

Assume S the set of all natural numbers that can't be described with 14 words or less (which is not an empty set). This set must have a minimum natural number N because:

Let another number X of the same set.

If X<N then there is a smallest number (it's just not N)
If X>=N then N is the smallest number

This is a valid proof? Because it seems to me like yet another attempt to apply the concept of infinity to a system designed only to work with finite numbers.

What is wrong with the proof? I am not saying that this is the most elegant proof you can find, but it's valid.
What infinity has to do with it?

A practical way to see that there is a smallest natural number N is the following:
does 1 belong to the set S? NO? Then
does 2 belong to the set S? No? Then
................................................
does N belong to the set S? Finally the first yes? Then this is your smallest number.

[Bridge]

My point is there is no such thing as a "smallest" number that is measurable. We use negative infinity to approximate it because there is absolutely no way to know what it is, and infinity doesn't really mean anything on its own. Set theory is designed to work with finite numbers and the infinity abstract, but it doesn't take into account an infinitely small measurable number because it was not designed that way (and it is impossible). Therefore it is impossible to conclude there is definitely such a thing as a "smallest number" using a system that doesn't know what that is.

EDIT: To further illustrate, the reason why the proof tells you there is a smallest number is because it expects it. I do not believe we have sufficient evidence at this point to ascertain if infinity can be followed through to a definite, finite number and therefore we cannot use the tools we made to say that there is. We made it to work with finite numbers - it doesn't work with unknowables.

[BAndrew]

My point is there is no such thing as a "smallest" number that is measurable. We use negative infinity to approximate it because there is absolutely no way to know what it is, and infinity doesn't really mean anything on its own. Set theory is designed to work with finite numbers and the infinity abstract, but it doesn't take into account an infinitely small measurable number because it was not designed that way (and it is impossible). Therefore it is impossible to conclude there is definitely such a thing as a "smallest number" using a system that doesn't know what that is.

Wait. We are talking about Natural numbers.

This is the set
N{1,2,3,4,5.....} and maybe 0 depending on who you talk to.

No negative numbers are on this set and no irrationals (eg sqrt(2))


Note: The smallest Real number doesn't exist

Proof:

Spoiler below!

Let n the smallest real number
n-1 is a real number and smaller than n
contradiction

And that's why we use -∞

[Bridge]

Wait. We are talking about Natural numbers.

Haha, I'm sorry. I read "integers" and my mind just crossed over the natural part. Well then I suppose I'll need to read the problem and thread over again before I can hope to contribute something meaningful to this thread.

[Froge]

To say that the statement "a number N cannot be described with 14 words or less" is a description of that number N is self-contradictory.(1) It's like


the sentence below is true
the above sentence is false
(2)

I didn't understand why (1) is similar to (2)

(1)

Proposition: a number N cannot be described with 14 words or less

Conjecture: The proposition is a way of describing N with <= 14 words

Paradox 1: But according to the proposition, N cannot be described with <= 14 words

Paradox 2: But according to the conjecture, which was based on the idea that the proposition is true, N was just described with <= 14 words

Paradox 3: Same as paradox 1

Paradox 4: Same as paradox 2

and so on

Which is the same type of circular going back and forth as (2)

[BAndrew]

To say that the statement "a number N cannot be described with 14 words or less" is a description of that number N is self-contradictory.(1) It's like


the sentence below is true
the above sentence is false
(2)

I didn't understand why (1) is similar to (2)

(1)

Proposition: a number N cannot be described with 14 words or less

Conjecture: The proposition is a way of describing N with <= 14 words

Paradox 1: But according to the proposition, N cannot be described with <= 14 words

Paradox 2: But according to the conjecture, which was based on the idea that the proposition is true, N was just described with <= 14 words

Paradox 3: Same as paradox 1

Paradox 4: Same as paradox 2

and so on

Which is the same type of circular going back and forth as (2)

I now understand your logic. But why "the proposition is a way of describing N with <= 14 words" is a conjecture?