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/sci/ - Science & Math

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2524331 No.2524331 [Reply] [Original]

Are there infinite real numbers between integers?

If so, how is it that we can give bounded magnitude to things using numbers? Wouldn't with n length actually have infinite real length?

>> No.2524341

>>2524331

There's a difference between infinite length and infinite subdivisions of a length's measure. See also zeno's paradox.

Also in the physical world, there is a quantum of length.

>> No.2524337

something*

>> No.2524345

>>2524331
there are an infinite quantity of finite numbers between, say, 0 and 1

>> No.2524346

>>2524341
What is that quantum? And what would be infinite subdivisions?

>> No.2524350

>Are there infinite real numbers between integers?
Yes.
The set of integers is countably infinite. But the reals are uncountably infinite. Between every two integers there is an uncountably infinite set of real numbers. This can be rigorously proven, actually, though the rigorous proofs probably can't be fit into threads on 4chan very well.

>> No.2524353

>>2524346

http://en.wikipedia.org/wiki/Planck_length

>> No.2524354

>>2524346
There is no known quantum of distance. That's just speculation.

>> No.2524356

>>2524341
> quantum length

speculation

>> No.2524362

Why would it have infinite length? an object of n length is n length if you measure it i ns, or n/2 s or n/3 s or n/100000000000000 s
It doesn't matter how small your unit is, it will still be n length

>> No.2524364

>>2524350
This.

The cardinality of the set consisting of all integers (aleph null) to the power of itself = the cardinality of the set consisting of all reals.

Basically aleph null ^ aleph null = the set of all reals.

I would do it in jsmath but I'm retarded and don't know how.

>> No.2524363
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2524363

There are an infinite number of rationals between two integers
There are no more rationals than integers
Both this statements are true in the current axiom set
Sorry for BLOWING YOUR MIND

>> No.2524361

>>2524356
Exactly.

>> No.2524365
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2524365

so wait. You can have infinite subdivisions but not be infinitely large? That doesn't make sense.

>> No.2524366

>>2524350
diagonalisation proof of uncountably of reals between 0 and 1 would probably fit in a single post

>> No.2524367

>>2524350
cantors diagonal argument should be possible.

>> No.2524373

hey did you ever look at an object an see it be infinitely long? thats fucking why you god-damned idiot

math just explains some aspects of reality, MATH DOES NOT CREATE REALITY TO FIT ITS OWN IMAGINATION

>> No.2524372

Almost all positive integers are larger than Graham's number.

>> No.2524371

>>2524350
>>2524350

This. Thread's over.

>This can be rigorously proven, actually, though the rigorous proofs probably can't be fit into threads on 4chan very well.

If any of you are wondering what he means, he's referencing a denumerability chart. The chart allows us to see if a set(say, N) is countably infinite or uncountably infinite. The proof he's talking about showcases an issue with the diagonal aspect of the chart proof. A popular proof showcases that [0, 1] is incompatible with the natural numbers or N. As such, uncountably infinite.

>> No.2524376

>>2524365
yes it does. Because those infinite subdivisions are infinitely small. If anything, it should have 0 length.

>> No.2524379

>>2524365
surely you accept none of these numbers are infinitely large...

1/2, 1/3, 1/4, 1/5, .... etc

...as they're all less than 1

and they all occur between 0 and 1, yet there are an infinity of them.

if you don't accept, then troll

>> No.2524378

>>2524376
an infinite value divided by any non zero would still have infinite magnitude for each subdivision.

>> No.2524380

>>2524365
These infinitely many numbers or separated by infinietly small gaps. Don't know if it will make more sense for you that way.

>> No.2524384

>>2524378
but the object isn't infintely large, so you aren't starting with an infinite value

>> No.2524386

>>2524379
but something divided by infinite.......

>> No.2524394

>>2524376
>>2524380
No such thing as "infinitely small" in the context of standard real numbers. Look at >>2524379; the gap between each pair of numbers is finite.

>> No.2524396

>>2524384
but you're dividing it infinitely. Which renders it infinite.

>> No.2524401

>>2524386
lim 1/x=0
x->+∞

>> No.2524402

>>2524350
This leads to another fun fact: Almost all numbers are not computable. This means that no algorithm, of any finite length whatsever, with any arbitrarily large amount of memory and time, can return those numbers as its answer.

Here's a simple proof:
The set of integers is countably infinite.
All programs (implementations of algorithms) can be expressed by a single integer in a Turing machine.
Each program can only return one result
Thus, the set of computable numbers has the same cardinality as the integers.
The set of integers is countably infinite, and so is the set of computable numbers.
But the reals are uncountably infinite.

Therefore, almost all numbers are incomputable.

The conclusion is right, but feel free to correct me if I've made mistakes in the argument.

>> No.2524403

>>2524379
>infinity of them.

that would mean the sum of the divisor is infinite. What's n divided by infinite?

>> No.2524407

>>2524394 here,
... except for zero, I should add. But two numbers separated by zero distance are the same number.

>> No.2524410

>>2524396
why does it?

>> No.2524406

>>2524378
Division is a function of two real variables, infinity is not a real number.

>> No.2524419

>>2524403
You're not really doing division here because the lengths you're dividing the interval into aren't all the same.

>> No.2524420

>>2524386
nope

only divided by all the natural numbers

next you'll be claiming there is an integer called infinty because the integers are infinite

lrn2 distinguish an infinite set from an infinite cardinality

anyway troll

>> No.2524426
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2524426

>> No.2524459

they are floating points, not integers

>> No.2524489

>>2524459
>floating points
CS Major detected

>> No.2524502

>>2524356
So you are saying that no length can ever be measured, and is never quantified?

That's great to know! I'll definitely use that one the next time I measure my dick!

>> No.2524526

>>2524502
It looks like all he's saying is that length isn't quantified. Which it isn't. How do you arrive at your conclusion from that?

>> No.2524534

There is an infinite amount of subdivions, each of which are infinitely small. Infinity x 1/infinity is not (always) infinitely large.

>> No.2525151

>>2524526
How the fuck is length not quantified? 1 meter is 1 meter no matter how the fuck you look at it.

>> No.2525203

>>2525151
It's quite obvious that "quantized" is what was meant.