> Yes, but the list can be given by a formula only. I did not exclude > that. I said: A real number can be listed by a terminating decimal, by > a periodic > decimal, or by a formula supplying each of its decimals. > It is not possible to list irrationals *by writing their decimals*.
I never claimed it was possible, nor was it necessary. A formula still determines each decimal in a unique way, just as writing the decimals would.
> A formula is not writing the decimals.
A formula allows me do get whichever decimals I need for what I'm doing , and deduce properties about them in general . A formula is partly like a vending machine (with infinite storage) , and the n'th decimal is like the n'th type of soda. When I order the n'th decimal , I don't expect the vending machine to spew out all the sodas like crazy (write down all digits) . I just want my n'th soda. Nothing more , nothing less. When I need the m'th soda , I'll order the m'th soda .
> > The most accepted cosmological model predicts the universe is > > spatially infinite and roughly homogenous in large scales. > > The realm we can ever use is finite.
Even if that were true, scientists use observations about this 'finite realm' to predict that the universe as a whole must be infinite.
> Entropy is increased elsewhere. > > > Information is never truly 'erased' . > > Switch off your pocket calculator. > > Regards, WM
A highly naive answer, coming from a 'physicist'. 'Erased information' is simply transferred in the environment, in a form such that it's recuperation would be impractical to us (it no longer appears as information to us , but as 'heat' ) . It's never erased . http://en.wikipedia.org/wiki/Landauer%27s_principle
Let's reiterate how a proof by contradiction works :
Proving forall y , (non (P(y)) : You claim to have x such that P(x) holds . I take x , and generate a counterexample to your claim (thus proving it false) .
Then, you claim to have z such that P(z) holds . I take z , and generate a counterexample to your claim (thus proving it false) .
.................................... If my counterexample-generating procedure is universal (I can refute all your possible claims ) , then I win , but the example-refutation game can go on forever . forall y , (non (P(y)) is proved .
On the other hand , if you claim , P(k) holds for some exemplified k , and I cannot generate a counterexample , then you win . forall y , (non (P(y)) is disproved . exist k , P(k) is proved .
Now then, let's see how the Cantor game works : You claim that the reals (between 0 and 1 for example) are denumerable . Therefore , you must present a list L_1 (or a formula for a list ) than contains all the reals . "exists L_1 , contains_all_reals( L_1)"
But, I take your list L_1 , and use the antidiagonal to construct a real number R_1 , that is not in the list .
You can then build a better list , L_2 , that contains all the members if L_1 , in addition to R_1 , the counterexample for L_1 .
But, again, I take your list L_2 , and use the antidiagonal to construct a real number R_2 , that is neither in L_2, nor in L_1 .
And the game goes on forever .
This means I win . There is no list (or formula for a list) that contains all real numbers , therefore , I dub the real numbers 'un- listable' .