Let's suppose there exist an infinite number of positive integers.
Let's consider all the postive integers that can be defined in the english language with a sentence of less than 1000 characters.
A sentence is a definition of a number if one and only one number correspond to the sentence. (not all sentences define a number)
It is easy to see that there is a finite number of sentences of less than 1000 characters, since there is an infinite number of positive integers, obviously some numbers can't be defined with less than 1000 characters.
Let's call N the smallest of this numbers that can't be defined with less than 1000 characters.
N can't be defined withh less than 1000 characters, yet the sentence:
-"The smallest number that can't be defined with less than 1000 characters"
contains less than 1000 characters and define N.
Hence N can be defined with less than 1000 characters, by contradiction there exist only a finite number of positive integers
Here I proved maths is just a meme. Can we move on now and stop pretending math is important.
I guess you're trolling, but just in case
https://en.wikipedia.org/wiki/Berry_paradox
>>7914747
You have not proven N exists, so the proof does not work.
>>7914831
Every subset of the positive integers contain a minimal element.
Even my 6 year old daughter know that.
>>7914817
So someone proved maths was just a meme before me, what's your point ?
>>7914747
>contains less than 1000 characters and define N.
It doesn't define it, nor are you using a consistent definition of character: I supposed you were using 0123456789, and now you seemed to refer to 1...0,A...z. I don't know the value that you are assigning to each character neither. So you clearly haven't defined what you mean by "character", thus you haven't define N. Sucks to be a fucking retard, I guess
>>7914893
That doesn't respond to my point. What does "can't be defined" or "can be defined" mean?
Let's consider all positive integers which are flibble
Then N is the smallest positive integer which is not flibble
But N is flibble
Therefore math is wrong.
>>7914931
You're obviously retarded. I will report your post, in the hope I don't see anything that stupid again
>>7914931
pic related
>>7914943
20 can be defined by "twenty" or "10+10" or " the integer between 19 and 21" or "my dick's length in cm" .
Any rigorous mathematical definition of twenty that can be written in less than 1000 ascii characters is a definition of twenty that contain less than 1000 characters.
I don't have any example of numbers that can't be defined in less than 1000 characters, but the fact they exist is obvious, since AT MOST 128^1000 sentences of less than 1000 characters exist, so AT MOST 128^1000 positive integers can be defined with such sentences.
>>7914901
better question is what is your fucking point
>>7915024
Math is as rigorous as feminist science.
>>7915002
>I don't have any example of numbers that can't be defined in less than 1000 characters, but the fact they exist is obvious, since AT MOST 128^1000 sentences of less than 1000 characters exist, so AT MOST 128^1000 positive integers can be defined with such sentences.
This assumes that the concept makes sense in the first place. What the paradox shows is that it is impossible to calculate how many characters are required to "define" a number.
>>7915040
I'm pretty sure we could lock a mathematician for a few gorillions years that would go through every possible of the 128^1000 sentences and decide which define a number and which don't.
>>7915061
That would require a rigorous definition of what it means for a sentence to define a number. Either this definition would allow this recursion and therefore fail the criteria, or it would not allow recursion and thus "the smallest of this numbers that can't be defined with less than 1000 characters" would not be a definition.
You can't formalize your proof because "defining numbers with characters" is not a well-defined concept in ZFC.
So you're wrong
before you all go on, define "define".
>>7915071
so the problem would be coming from the recursion ?
>>7914747
>obviously some numbers can't be defined with less than 1000 characters.
wrong.
it depends on which base you express your number with.
to base N the number N is always 10
>>7915173
you wot m8
>>7914747
Let's take this sentence and count characters: "The smallest number that can't be defined with less than 71 characters"
3+8+6+4+5+2+7+4+4+4+2+10+11 = 70
Is this sentence true? Does the smallest number that can't be defined with 70 characters exists? Then the number is not defined in this
sentence, thus the sentence doesn't define it.
Is this sentence false? Does the smallest number that can't be defined with 70 characters not exist? Then the number is not defined in this sentence, thus the sentence doesn't define it.
In either case, the number is not defined, as expected.
>>7914747
>Let's suppose there exist an infinite number of positive integers.
The number of positive integers defines an infinity not the other way around.
>Let's consider all the postive integers that can be defined in the english language with a sentence of less than 1000 characters.
You have set a condition that at minimum simply selects a subset of positive numbers from the start.
>A sentence is a definition of a number if one and only one number correspond to the sentence. (not all sentences define a number)
An unnecessary condition. It simply states that each sentence that defines a number uniquely defines that number. Again a condition that simply narrows any numbers not excluded by your first.
I am not going to opine on your proof but even if valid all you have proven it for is true for the numbers satisfying your selection criteria.
Counter proof (I am going to keep it simple):
- Given your defined set of finite number of positive numbers.
- These can be shown to be well ordered (i.e. for any two a and b either a>b or b>a)
- Any finite set of well ordered elements has a maximal element.
- Add one to the maximal element.
- These creates an element that was not in your original set (or it would have been the maximal element).
- Proof by contradiction that the positive numbers are not finite.
- If they are not finite, we can define the concept of infinity and it's properties will depend on the set and set properties of the system we are studying.
>>7914747
Argumentation is right, but old, and ZFC now exists
>>7915415
What the fuck are you talking about, you psychotic retard
>>7914831
>>7914931
>>7914943
>>7915040
>>7915173
>>7915236
>>7915415
I'm almost certain you are all wrong.
Let's recap the problem with sentences of less than 2 characters.
There are 128 sentences if we consider ascii char, only 10 define a number by any rigorous mathematical standard:
0
1
2
3
4
5
6
7
8
9
The concept "define" is perfectly well defined, the smallest number that can't be defined with a sentence of less than 2 characters is 10.
The only anon who seems to get what's the problem is >>7915071
there is no problem with 1 character or a few characters because we can't make recursive definition wich seem to fuck the problem up, but I have no idea why.
>>7915539
for sentences with less than 3 characters, I think the biggest number you can define is "9!", but the smallest you can't define is 100, but I'm not sure, I may have missed a possibility.
