Prove convincingly to me that the real numbers exist. Come on,

look at your hand, it has 5 fingers, so 5 exists. there's then no reason for all the other numbers to not exist

Let 1 be a real number. If we let x = 1 and x is in Z, then QED

>>8323474
Assume that only a single number exists, say 1.
We know that 1 = 1
Assuming that 1 exists add 1.
1 + 1 = 2, a newly defined number
We proved that any x + 1 = a new number
this holds for all cases
QED

>>8323493
Why assume 1 exists?

>>8323494
We must assume something for an inductive hypothesis, as long as this hypothesis does not encounter a contradiction it is true.

the fact that I was able to derive a conclusion from my "made up" hypothesis shows that its true

>>8323474

Define "exist".

>>8323474
Stop trolling /sci/ asperger's guys!!

>>8323474
Numbers exist as much as the distinction between animate matter and inanimate matter exists; they are useful abstractions

>>8323477
>>8323481
>>8323493
> ask about real numbers
> morons try to prove the existence of naturals

>>8323513
1 + 1 = 2
"half of 2" = 1
half of 1 = ???
>Conduct another inductive hypothesis to prove 1/x, similarily you prove other forms like this as well

God, you think youd think people would at least try to think before they post?

>>8323474
They don't exist. They are an artificial construct which has been shown to be consistent with widely used axiomatic systems and laws of deduction.

>>8323550
Define """exist"""?

>>8323474
My Real Analysis lecturer already did this

>>8323474
"Nothing" is a self evident concept. "Something" is also a self evident concept. It is clear that you can have more or less of "something".

As you can have different amounts of something, it is easy to see that there is a smallest possible amount of "something" you can have. We call this 1. We can extend this to see that there is a smallest possible amount more we can have. We call this 2. Repeat ad infinitum

>>8323521
> goes on to show a couple rationals
> still doesn't know what a real number is

>>8323474
Assume the integers and rationals exist.
Let:
A sequence be a function that maps positive integers to rationals.
A Cauchy sequence be a sequence such that
[eqn] \forall \varepsilon \in \mathbb{Q}^+ : \exists N \in \mathbb{Z}^+ : \forall m,n \in \mathbb{Z}^+: |s_{N+m} - s_{N+n} | < \varepsilon [/eqn]
Let [math] \mathcal{F} [/math] be the set of all Cauchy sequences.
[math] \mathcal{F} [/math] is clearly not empty.
A sequence is a null sequence if
[eqn] \forall \varepsilon \in \mathbb{Q}^+: \exists N \in \mathbb{Z}^+ : \forall n \in \mathbb{Z}^+ : | s_{N+n} | < \varepsilon [/eqn]
Let [math] \mathcal{Z} [/math] be the set of all null sequences.
Define subtraction on sequences in the obvious way: {s} - {t} = {s-t}
Two sequences are equal mod [math] \mathcal{Z} [/math] by
[math] {s} \equiv_ {\mathcal{Z}} {t} [/math] iff [math] s - t \in \mathcal{Z} [/math]

Let the reals be the set of equivalence classes defined by [math] \equiv _ \mathcal{Z} [/math]

I don't think reals exist in real life the same way they do in maths. You can't have an infentesimal slice of cake because at some point you would be cutting a slice smaller than an atom...

They don't exist. Take a segment of the number line with length L. L contains an uncountably infinite number of points which represent the real numbers. Remove a point from the line, and neither the length of the segment nor the number of points changes. You can do the same for a countably infinite number of points and not affect the line in any way. You cannot remove something from reality and have reality be exactly the same; therefore, the real numbers are actually "nothing" in that they do not exist.

You could argue that the line is not exactly the same because it is missing the elements which were removed. This is an incorrect view of the situation. Imagine I remove all of the unicorns from the universe (I mean actual unicorns and not thoughts or artistic depictions of them). The universe is unchanged because they never existed in the first place. A single element of an uncountably infinite set (the point in a line) does not exist in the same way the unicorn does not exist: a point has no length and a unicorn has no coordinates in spacetime.

Going back to the original example, if I were to remove (1/3)L from the line, it is now clearly changed. Lengths are real in this example because we have already defined the line as existing. Points do not exist with respect to the line because they are not of the same dimension. Similarly, planes do not exist in 3D space because their absence is not felt. Our universe has 3 dimensions of space and one of time in a possibly simplified view. To exist in our universe, one must be have three spatial dimensions and one temporal dimension; real numbers, represented geometrically as points, do not meet the criteria.

>>8323497
>believing in induction
>induction works because it's always worked, guys!

kek

>>8323679
It's like you never even read the domino analogy

>Prove convincingly to me that I exist.
You do not exist.
QED

>>8323474
What do you mean with 'exist'? The physical manifestation of a real number in the external world? Something else? The definition of real numbers exist, ergo, the real numbers exist at least axiomatically.

>>8323481
kekek

prove that any number exists

oh wait you can't

guess I'll keep messing around with my dirty dirty reals