Try to disprove that ZFC is coherent within the possibilities of ZFC.
Pro-tip : you can't.
>ITT : we pretend to have broken mathematics.
Congrats on reading Godel OP, he's cool and smart. The next step to enlightenment is realizing that math and science is nebulous and socially constructed - which doesn't mean it isn't real in some sense, money is a social construct and it definitely exists, but it isn't some perfect crystalline edifice of pure reason.
>>8007217
Maybe you should read Foucault's "The order of things" and a bit of Popper. (Reasonable) Modern science doesn't claim being an immovable monolith of pure truth.
>>8007238
I have read Foucault! He's a very smart dude.
Unfortunately scientism is really embedded in modern culture and rationalist epistemologies are far from uncommon.
>>8007210
[eqn] \sqrt{-1}=\sqrt{-1} [/eqn]
[eqn] \sqrt{\frac{1}{-1}}=\sqrt{\frac{-1}{1}} [/eqn]
[eqn] \frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}} [/eqn]
[eqn] i=\frac{1}{i}=-i [/eqn]
>>8007268
In the second line you just defined i to be its reciprocal.
>>8007268
sqrt(-1) is not defined. (i is defined as i^2 = -1)
>>8007279
How about we define [math] \sqrt{x} [/math] as being the root for which [math] \Re(z)>0 [/math] if [math] \Im(z)=0 [/math] and the root for which [math] \Im(z)>0 [/math] if not ?
>>8007291
where did z come from? I think you're referring to the principal square root. The normal algebraic properties of the square root function for nonnegative numbers does not hold for the principal square root, so for complex numbers a and b, it is not generally true that
[eqn]
\sqrt{ab} \neq \sqrt{a}\sqrt{b}
[\eqn]
>>8007217
>nebulous and socially constructed
>real in some sense
>crystalline ediface of pure reason
Lmao, this board blows. Goddamn philofags
>>8008245
>implying axiomatic set theory and the concept of infinity is not nebulous
Math noob here, why can't I just disprove the continuum hypothesis by constructing a set whose cardinality is [math]\frac{\aleph_0 + \aleph_1}{2}[/math] ?
>>>Axsom of foundation<<<
A member string ([math]A \,\in\, B \,\in\, \cdots[/math]) cannot be infinite.
>>>Axiom of infinity<<<
There exists [math]A[/math] such that [math]\varnothing \,\in\, A[/math] and [math]A \,\cup\, \left\{A\right\} \,in\, A[/math] which entails an infinite member string.
uhh
[math] (...\in A \in B \in C) [/math]
>>8009151
How is it any different, enlightenedfriend?
>>8009182
The axiom of Infinity doesn't violate that one.
Pick any set. It can be [math]\mathbb{N}[/math] for instance. According to you it violates foundation because:
[math] 1 \in 2 \in 3 \in ... [/math]
and it violates what you wrote. But it doesn't violate what I wrote. Take any number. You can't construct an infinite downwards chain, because there's only finite things below.
Hell, take any ordinal, no matter how it is. Aleph0, AlephAleph, whatever. You can't have an infinite downwards chain even if there's an infinite amount of things below you because every time you take a downwards step, you're forced to make huge jumps.
>>8009151
>because you can't construct that
well that's me convinced
>>8009253
Infinite cardinals don’t work like real numbers.
[eqn]\frac{\aleph_0 \,+\, \aleph_1}{2} \,=\, \frac{\aleph_1}{2} \,=\, \aleph_1[/eqn]
>>8009344
>Implying you can't build a bijection between [math] \mathbb{R} [/math] and [math] \mathbb{R} - \left\{0\right\} [/math]
Spyware is this thread BE CAREFUL guys plz
>>8009300
>divided by 2
That is not defined in this case. There is no ordinal [math] \alpha [/math] such that [math] 2 \alpha = \aleph_1 [/math].
>>8010109
yes, it's precisely aleph1 itself
>>8010132
Wait, so you're dealing cardinal arithmetic arithmetic then.
In which case division makes no sense.
Whether you're thinking of it as ordinal division or cardinal "division", you can't do it.
>>8010145
>There is no ordinal α such that 2α=ℵ1.
>yes there is, it's obviously ℵ1
>w-wait no, you must be wrong you can't do it!
gee you're really a faggot. it's ordinal arithmetic. 2ℵ1 = ℵ1.
you clearly can't define division as 2*ℵ1 != ℵ1*2 but what you said was wrong
>>8010147
If we're dealing with ordinal arithmetic, as I originally thought we were, to be clear let's use [math]\omega_1[/math] instead of [math]\aleph_1[/math].
Now, [math]2 \omega_1 > \omega_1 [/math]. They are not equal. If you do not think this, you do not understand ordinal arithmetic.
>>8010162
2ω1 is precisely ω1
ω1 2 > ω1 though
>>8010166
No. You are literally, objectively wrong.
I may be taking bait, because I have difficulty believing someone could be so confident in themselves when they are absolutely, completely wrong.
[math] 2 \omega_1 = \omega_1 + \omega+1 [/math] is, visually, two copies of [math]\omega_1 [/math] concatenated together. [math] \omega_1 < \omega_1 + 1 < \omega_1 + 2 < \ldots < \omega_1 + \omega < \omega_1 + \omega + 1 < \ldots [/math], all the way up to [math] \omega_1 + \omega_1 = 2 \omega_1 [/math].
>>8010173
you're thinking of ω1 * 2 friend
2 * ω1 is ω1 copies of 2 and that's ω1
>>8010175
Oh my god I'm retarded. I thought you had it backward, but it was I.
>>8010195
lol silly goose ;)
>>8010199
Still, when trying to define something like "[math]\omega_1 / 2 [/math]", it should clearly be the case that [math] \alpha / 2 [/math] should be the unique ordinal [math] \beta [/math] such that [math] \beta + \beta = \alpha [/math], if such an ordinal exists.
So [math] \omega_1 / 2 [/math] is undefined.
>>8009083
It would be better if you've read up on ordinal numbers and some set theory. But you can say that [math]\aleph_0 + \aleph_1 = \aleph_1 [/math].
And it's been a long time since I've actually studied logic and set theory, but I don't remember ordinal division being defined there, so there's that.
buewon
>>8007238
Science isn't a thinking entity that claims somethig.
Most phsicist (not get started with the other scientists and egn.) are naive realists and most matematicans couldn't tell you what modal logic is
when did /sci/ develope an interest for ordinal arithmetic?
