You don't. As you mention, all that matters is that each program, or machine, can be encoded by a distinct integer.
In computability theory, the exact Gödel numbering isn't particularly important, it's much more the concept itself that matters.
(Notice the umlaut over the 'o' i Gödel, btw.)
Hi all,
Tim Gowers has a list of possible Polymath projects up, where he mentions the Littlewood Conjecture: This states, informally, that all reals can be approximated reasonably well, in the sense that if we let the denominator of an approximation \frac{u}{v} grow, the error term becomes...
Remember that measurement is projection and renormalization.
Alright, so |\psi\rangle=a_{00}|00\rangle+a_{01}|01\rangle + a_{10}|10\rangle + a_{11}|11\rangle. A measurement of the first qubit as 1 means that |\psi\rangle should be projected onto the subspace spanned by |10\rangle and...
The reason I ask is that I recall an argument that if Fermat's Last Theorem was undecidable, then it would automatically be true. In ran along the lines that if it couldn't be proven false, then no counterexamples could exist. This certainly seems reasonable, so it was concluded that if FLT was...
Is undecidability strong enough to give us truth/falsehood, even though we cannot prove it inside any given theory?
Consider the continuum hypothesis. It is undecidable, so, clearly, there's no counterexample of a set with cardinality between the integers and the reals (in standard ZF)...
If you haven't already, I suggest you get Anders Thorup's "Algebra" from Naturfagsbogladen which lies in August Krogh Institutet on Jagtvej. It's cheap (roughly 120 kr. when I bought it) and is used as a second year introduction to algebra on Københavns Universitet.
It's an excellent...
Can you see things that do not have the same time-cordinates as yourself in your own frame of reference (not taking into account the time it takes time to travel from them to you)?
Well, people tend to view congruence defined as a ternary relation:
x \equiv y \mod n \overset{def}{\Longleftrightarrow} n|x-y.
Sometimes one omits the modulo part, but it is still understood that we're dealing with modulo arithmetic by using the equivalence sign \equiv, instead of an...
That Escultura guy looks like a real crackpot. I was meserised at his http://www.manilatimes.net/national/2004/aug/28/yehey/opinion/20040828opi4.html about 1\neq0.99\ldots Even if one takes his babble at face value, and take the reals to be constructable infinite sequences of naturals, ordered...
As to why the infinity of the amount of rationals doesn't imply that every number is rational:
The infinity of the number of reals is bigger than that of the number of rationals, |\mathbb{Q}|<|\mathbb{R}|.
So even though the rationals are an infinite set and a subset of the reals, they can...