prove me wrong (or right)
"red equals" = f(x) = SQRT(X^2)
"green equals" = g(x) = 4 - 2x.
(in both cases, "x" refers to the value of the expression on the left of the equals operator)
You're wrong,
unless you're sloppy and by '1+1' you mean 'the number after 1', for which '2' is an abbreviation and then the second line is not the logical equality.
In a predicate logic with equality '=', you have
[math]\forall x.\, x=x[/math]
by definition of '='
and thus, for every b in your domain of discouse,
[math]b=b[/math]
as an almost trivial theorem.
If you set up the natural numbers, usually you say 1 as short for S0, where S and 0 are actually primitive, and then 2 for S1. Let us write this as
1:=S0
2:=S1
Then you set up addition '+' recursively via pic related, so that
1+1
which is abbreviation for
S0+S0
reduces to
S(S0+0)
which reduces to
S(S0)
which is
S1
which is
2
Thus
1+1=2 is a theorem in this theory.
Again, if you're sloppy and write n+1 for Sn (which is provably equal), then you may write the definition of 2 as 2:=1+1 and then they would not be equal.
>>7722881
Here is an overview of the axioms, but of course there are many formulations of this basic structure
A cool constructive one involving mostly universal constructions is
https://en.wikipedia.org/wiki/Initial_algebra#Example
(but I just mention it because it's sweet and generalizes to so many other data structures.)
>>7722845
Not even sure what your point is, however I am sure that you're wrong.
How do you guys deal with the fact that the numbers that describe our reality are constantly changing and fluctuating and every single measurement you could ever take becomes obsolete as soon as you finish making it?
If even the atoms that make up matter are in a state of constant movement and flux, is there any such thing as non abstract mathematics?
Essentially, solving a math equation is no different than winning a video game. You may have satisfied some arbitrary parameters, but neither the parameters nor your satisfaction of them correspond in any way to reality.
Do you guys have a philosophical take on this?
>>7722845
define "="
>>7722939
>Do you guys have a philosophical take on this?
Yes, that nobody who's interested in math should be a fucking Platonist (at the weekends)
http://plato.stanford.edu/entries/platonism-mathematics/
>>7722845
>tfw colorblind
autistic post of the year.
>>7722913
Wait a second
doesn't 7. follow from 1.?
If you can have no number that is followed by 0, then no number can be smaller than zero thus no number can be negative
or is there some special case where it does not follow?
>>7722968
S(-2) != 0 so x= -2 satisfies axiom 1 while not satisfying 7. Conversely, it seems like 7 and 8 imply 1 though.
Assume: there exists x: (Sx=0)
for all x: x=x
for all x: (x < Sx) (follows from the previous statement and axiom 8)
because there exists an x such that Sx = 0, call that x n
Sn = 0
n < Sn (follows from Sn=0 and for all x: x<Sx)
n < 0 (follows from Sn = 0 and n < Sn)
for all x, not x < 0 (axiom 7)
contradiction
there does not exist x: (Sx=0), which is equivalent to saying for all x: not(Sx=0)
>>7722968
I haven't taken linear algebra or number theory or anything like that but intuitively, if no number can can be followed by 0, it doesn't mean that it can't be preceded by a number greater than 0. In other words, there is no number that will give you a different number if you add or subtract 0 to or from it, but if you add or subtract a number greater then 0 to or from it, then you will get a different number, which is consistent with basic arithmetic.
>>7722845
Simple: 'equals' is a concept, not a value. That's like saying infinity is a number; it's not.
>>7722967
KEK "and the winner is?"
>>7722966
Really? What color is this!? What color is this, anon!?
>>7723386
You forgot to redtext.
- Red equal is any reflexive binary relation on the integers,
- Green equal is any other reflexive binary relation on the integers.
Done.
>>7723018
Yeah, I don't see anything wrong with your prove (you could write it down shorter), but the axiomatization in the pic also doesn't coincide directly with the PA and Robinson arithmetic axiom in the pic.
Maybe it's not minimal, or maybe they wanted to keep it constructive and not have you switch \forall\not to \not\exists.
>>7722845
Could someone explain to me wtf is going on itt?
>>7722845
Sup terry
>>7722881
>he thinks [math]\forall x, x=x[/math] is an axiom
needs moar rigor
>>7723290
>this meme again
There are several branches of mathematics where treating infinity as a number makes things work out a lot nicer.
Quit being so goddamn undergrad.
>>7724380
literally the definition of equality.
first of all get your shit straight.
It should be =!==.
Thank you.
>>7724611
fuck you 4chan my name has an ! in it damn it!
>>7724406
No. Reflexivity follows from Leibnitz's law.
>>7722881
>unless you're sloppy and by '1+1' you mean 'the number after 1', for which '2'
That's literally the definition of sum
>>7724627
what?
For [φ] a (Gödel-coded) formula and s a variable assignment, let `Sat([φ],s)' abbreviate the following second-order formula
&forall R {
{for any (coded) formula [ψ] and any variable assignment t
(R( [ψ],t) ↔
( ([ψ] = `x_i ∈ x_j' ∧ t(x_1) ∈ t(x_j)) ∨
([ψ] = `x_i = x_j' ∧ t(x_1) = t(x_j)) ∨
([ψ] = `(∼θ)' ∧ ∼R([θ],t)) ∨
([ψ] = `(θ∧ξ)' ∧ R([θ],t) ∧ R([ξ],t)) ∨
([ψ] = `∃x_i (&theta)' and, for some an xi-variant t' of t, R([θ],t'))
)} →
R([φ],s)}
>>7724888
What's with all the unbounded formulas? ξ, etc.
Also, how does relate to the number 2?
>>7724383
>he didn't embrace our superior constructivist overlords
>>7724908
The smallest number bigger than every finite number m with the following property: there is a formula φ(x1) in the language of first-order set-theory (as presented in the definition of `Sat') with less than a googol symbols and x1 as its only free variable such that: (a) there is a variable assignment s assigning m to x1 such that Sat([φ(x1)],s), and (b) for any variable assignment t, if Sat([φ(x1)],t), then t assigns m to x1.
>>7724841
Suppose x and y. Say that x = y if and only if p(x) iff p(y) for all p, where p is a propositional function.
You get x = x from this.
1/1 = 1
1/2 = 2
===
there you go.
nature gives a shit about your philosophical take on this.
>>7724932
oops, should be
1/1=1
2/1=2
fuk me rite
equality does not exist
how do you say one apple = one apple? they are not the same size exactly nor the same color or same shape
even if you take two elementary particles they are not equal because they don't occupy the same quantum state
mathematical equality is just a meme mathematicians came up with
in reality math has no practical purposes
it's just an arbitrary game, like chess
when you do math you are literally wasting your time playing chess
>>7724940
You're retarded
>>7725018
define "you're"
where does the entity known as "you" begin and where does it end? Are you a part of me? how can you tell?
>>7724940
Despite what you write being a little confused, I guess it's worth pointing out that a lot of current mathematical research is all about providing a good theory of '='.
See e.g.
https://en.wikipedia.org/wiki/Homotopy_type_theory#Univalence_axiom
Also, what you said reminded me of this paper
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
>>7725148
that's because I wrote that paper
>>7725185
I'm trying to enlighten you by playing doubles advocate
think of me as the wise sage in a koan and yourself as the humble student
>>7725185
Even a single apple cannot be equal to itself because the first apple occurs earlier in time than the version of itself you are comparing it to
