0^0

ONE WORD, 0⁰ = ?, /sci/ over

>>40
booyah, you told him, Vinny

>>37

Frege and Russell took 0 to be { {} }, but von Neumann took 0 to be {} (and n+1 to be n U {n}), and this is quite standard in many contexts.  Your assertion about "almost every contemporary text" is just plain wrong; you obviously need to read more contemporary texts.

See
http://en.wikipedia.org/w/index.php?title=Ordinal_number
for more about the von Neumann 0 = {} thing.

>>38

You blew it: you forgot to consider that K could be {}, which satisfies your requirement. So the set of such K might be { {} }, and so have one element.

And in fact when S is empty, the set of functions from {} -> S is { {} } and there is this one function.

See:

http://en.wikipedia.org/w/index.php?title=Empty_function

>>42
As long as you're going to drag the conversation down to the realm of citing wikipedia, http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

>>37

Frege and Russell both defined 0 as { {} }. Furthermore, this is the definition used in almost every contemporary text. Please take set theory 101 before trying to have a conversation with the adults.

I can't say it any better than wikipedia does "This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theory ZFC and related systems, because in such systems the equivalence classes under equinumerosity are "too large" to be sets. For that matter, there is no universal set V in ZFC, under pain of the Russell paradox."

Or to prarphrase: it's informal; it's deprecated; it doesn't really work.

I would like you to name a couple of contemporary texts which use this definition and state that it is formal, because I don't believe you.

If you don't believe wikipedia and its citations then I direct you to http://mathworld.wolfram.com/OrdinalNumber.html .

>>38

You're logic breaks down the the case where S=E (ie. functions from the empty set to the empty set).

In this case the set ExS=E. There is one subset then of ExS, which is E. So let's set K=E and see if it is a map from E to S.

the condition you give is that "if (a,b) appears in K, then no other element of K is (a,c) for any c not equal to b" or in other words that "it is NOT TRUE that (there is some (a,b) and (a,c) both in K where b is not equal to c)".

Well, as K is empty, the bracketed statement is false, and so the entire statement is true; hence, by your own definition (which is indeed correct), the empty set IS a function from E to E.

Which is what I said in the first place.

Actually generally, by your definition again, there is ALWAYS exactly one function from the empty set to another set (the empty function). QED.

>>44

You're a dickhead. Everything you say is wrong. And you're sage bombing this thread because you can't handle it. GTFO and die.

>>47
Well gosh, I can't argue with that kind of logic!

X^0 is defined mathematically as X^y/X^y = X/X. Thus 0^0 = 0/0, but you cannot divide by 0, so 0^0 is undefined.

let g(0)=f(0)=0
lim(x->0) f(x)^g(x)=
lim(x->0) e^(ln(f(x)^g(x)))=
e^(lim(x->0) g(x)*ln(f(x))=
e^(lim(x->0) g(x)/(1/ln(f(x)))

L'hopital can take it from there.

>>49

No, indeterminate, not undefined.

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isnt anythign to the 0 power automatically 1?

Who honestly cares how you define the naturals.  All you need is one of these babies:
http://en.wikipedia.org/wiki/Natural_number_object

In a NNO 0^0 = 1

0⁰=0

>>50 wins.

I have a science project. I need to take a coffee can, and make it so that if the can is pushed forward it will roll forward and then roll backward to the starting point. what  do you suggest?

Tape a weight to the inside wall of the can.

i suggest pushing it up a hill.

ITT people with no real advanced maths education mix up really basic shit.

GET INSIDE THE CAN

SET YOUR HAIR ON FIRE

IT WON`T MAKE THE CAN MOVE, BUT IT WILL BE REALLY COOL

FIREWORKS, GASOLINE AND BLOODFLIES.
PUT THEM IN THE CAN.

ITT we divide by zero

By using Carls therm of convinces you will get 0^0=1

WTF????

>>28
Massive fail. See >>29.

Google says so. It must be true.

http://www.google.no/search?q=0%5E0&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a

can you apply L'Hospital's rule here or is this not one of the indeterminate forms?

>>70
see >>50

>>50

Fail.

>>56

Fail by proxy.

>>72
How does it fail?