Wikipedia:Reference desk/Archives/Mathematics/2008 December 14

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 14

Decidability of God

On what sets of axioms, if any, has the existence of God been proven undecidable? NeonMerlin 00:30, 14 December 2008 (UTC)

Can you express the existence of God in any appropriate language? --Tango (talk) 00:36, 14 December 2008 (UTC)

Well, you can use Bayes theorem (a famous theorem in probability) to calculate the probability that God exists. Now of course, you will have to 'make up' certain factors. So, take 10 events and based on each event, give your probability that God exists. From the totality of the 10 events, you can calculate the actual probability by going to the article linked. For example:

Event 1: There are earthquakes on Earth.

Conclusion: Does God have some hand in this? I guess 70% (this is not my actual guess but just an example).

Event 2: A particular experiment in physics has proven something illogical true (there are such wierd experiments in quantam mechanics).

Conclusion: Does God have some hand in this? I guess 90%.

Do this for a number of events (maybe 5 or so) and go to the article to work out how to calculate the probability of the existence of God. Of course, there are many assumptions involved in choosing these five events and then calculating the probability that God exists. In particular, the result won't be so accurate but it will be reasonably accurate (and give you are rough approximation) if you choose your events and probabilities wisely. I strongly encourage you to have a look at the article to ensure that you do so.

Topology Expert (talk) 09:33, 14 December 2008 (UTC)

Axiom: God is omnipotent.

Corollary: God can invalidate any logic one uses to decide anything about God.

Q.E.D. Cuddlyable3 (talk) 11:14, 14 December 2008 (UTC)

I have heard things such as, "If God can do anything, can he make a rock so big that not even he can lift it." But, the point in such thinking is, this is not an actual event. God can not do the impossible. On the other hand, his unlimited power allows him to do anything that is actually possible. An impossible event is not made possible by adding more power. By definition, it is impossible. (I did have an idea of what I had heard before but admit that I looked it up to say it better.) StatisticsMan (talk) 23:13, 14 December 2008 (UTC)

The OP didn't ask about the probability of the existence of God, they asked about the decidability. Those are completely different concepts. --Tango (talk) 14:43, 14 December 2008 (UTC)

Who cares? I told the OP something that might be of interest to him/her. Topology Expert (talk) 15:47, 14 December 2008 (UTC)

The biggest problem with a Bayesian analysis here is the difficulty of justifying your estimate of the prior probability that God exists. There are plenty of folks who will put that at 0, and plenty who will estimate it as 1, and in either case the posterior probability will equal the prior probability. --Trovatore (talk) 00:16, 15 December 2008 (UTC)

I have a theory about existence of God, that may possibly be of some help. I am convinced, with some good arguments indeed, that God exists, and does not exist, at the same time. This may sound contradictory at a first glance; but after a short reflection, you see there is no real contradiction (as shown e.g. by the theorem quoted by Cuddlyable3); clearly it looks paradoxical to us just because our mind is limited. But if you think, anybody is able to just exist, or, just not to exist: evidently God, in his/her omnipotence, is able to do both simultaneously. Of course I can't prove this as a theorem, but I wish to remark that this principle allows to explain quite easily any Religious Dogma, Sacred Mystery etc. (for instance: the Trinity Mystery, 3=1, become an exercise). Moreover, this principle embraces and welcomes all different points of view in matter of faith and religion, including the heretical and the schismatic ones, and even atheism, and in fact smooths their differences, which are a source of animosity and misunderstanding between men. I am, correspondingly, believer and not believer at the same time. A position very suitable for tolerance indeed, and I am delighted to confront it with any other one (who created the stars? God. When? Never. etc) --PMajer (talk) 13:57, 15 December 2008 (UTC)

Going back to the original problem, I might ask: in what set of axioms might one think the existence of God is not undecidable? That is, most axioms in mathematics or physics are different enough from any sort of God that one could probably make a larger system which is consistent with a God, and a different larger system which is consistent without a God.

Though, before one can actually try to figure out whether or not existence of God is decidable, one must rigorously define what that means. God usually is associated with the supernatural, and so perhaps existence of God should be equivalent to existence true statements which are not provable. Then Gödel's incompleteness theorem pretty much answers the question. Or perhaps existence of God should be equivalent to existence of something true inconsistent with the axioms. The latter makes more sense to me, but this is quickly becoming too much philosophy for me! GromXXVII (talk) 17:10, 15 December 2008 (UTC)

Does somebody really think that Gödel's theorems, and mathematical logic more generally, have something to do with phisical existence of whatever? Existence in mathematics and existence in the phisical word, are two different and well separated concepts, that we denoted with the same term just by a linguistic accident. Shifting from the logical to the ontological level this way is no more than making jokes. Unfortunately, it seems that Gödel himself, who suffered by mental disease in the last part of his life, contribuited to this misunderstanding, so that he is still quoted for proofs of statements about existence/non-existence of God. In my opinion, it's not fear towards a great mind like him to spread the few silly things that he might have said in the end of his life.--PMajer (talk) 14:55, 18 December 2008 (UTC)

While Gödel speaks specifically about mathematical logic I believe that Tarski's undefinability theorem can be extended to the "model" that is physical existence. Taemyr (talk) 05:43, 19 December 2008 (UTC)

Partial differencial equation

What is the transformatio in general form which can transfer the parial diffential equation u subscript xx + 4(u ubscript xy) + 4(u subscript yy) into canonical form. —Preceding unsigned comment added by Mr Sandeep mishra (talk • contribs) 05:46, 14 December 2008 (UTC)

what is transformation in general form which can transfer the partial diffencial equation u_xx+4u_xy+4u_yy=0 into canonical form. —Preceding unsigned comment added by Mr Sandeep mishra (talk • contribs) 05:51, 14 December 2008 (UTC)

Factor

$(\partial _{x}^{2}+4\partial _{x}\partial _{y}+4\partial _{y}^{2})=(\partial _{x}+2\partial _{y})^{2}$

The characteristics are lines 2x−y=C. A solution u has zero second derivative along the characteristics, and so solving this ode gives the general solution

$u(x,y)=f(x-{\tfrac {1}{2}}y)+\alpha y$

where f is an arbitrary function and α is a constant. siℓℓy rabbit (talk) 15:09, 14 December 2008 (UTC)

Fixed Points

How many fixed points does a linear fractional transformation have and why. —Preceding unsigned comment added by Mr Sandeep mishra (talk • contribs) 05:55, 14 December 2008 (UTC)

$y=1/x$

is a linear fractional transformation. It has one fixed point at $x=1$, a point at infinity where $x=0$ and a point at zero where $x=infinity$ Cuddlyable3 (talk) 11:20, 14 December 2008 (UTC)

Write down the formula for a general linear fractional transformation. Write down the equation for a fixed point of this formula. The result is a polynomial equation (ignoring singularities). How many solutions does it have? Fredrik Johansson 11:39, 14 December 2008 (UTC)

Depends on the domain and range of the transformation. Consider the following transformations:

$y={\frac {1}{x}}\quad ;\quad y={\frac {-1}{x}}\quad ;\quad y={\frac {x}{2}}$

first of all as transformations of the punctured real line $\mathbb {R} \setminus 0$ to itself, then as transformations of the punctured complex plane $\mathbb {C} \setminus 0$ to itself, and finally as transformations of the extended complex plane $\mathbb {C} \cup \{\infty \}$ to itself. First transformation has two fixed points in each of these domains, but the number of fixed points in the other two transformations depends on the chosen domain. Gandalf61 (talk) 10:57, 15 December 2008 (UTC)

Closures and cluster points

This is Problem 7.14 from Royden's Real Analysis. I want to make sure I am understanding this correctly.

Let E be a set in a metric space X. If x is a cluster point of a sequence from E, then $x\in {\bar {E}}$, while, if $x\in {\bar {E}}$, there is a sequence from E that converges to x.

Say E is a one-point set and x is the one point in it. Then x is in the closure of E. The book gives two definitions of cluster point. The first is that it is weaker than convergence and that we only require that each ball about x contains infinitely many terms of the sequence $\langle x_{n}\rangle$. I guess we form the sequence that has every term x in this case and then every ball about x has only one point in it but also infinitely many terms of the sequence. Does this count since every ball really only has 1 point (assuming the closure of E is just x)? I guess it makes more sense as I'm writing it here but it's still a little confusing. Thanks StatisticsMan (talk) 23:03, 14 December 2008 (UTC)

Yes, it counts. The sequence (0,0,0,0,0,0,0,0,...) of zeroes is a convergent sequence with limit zero. (lim_n→∞0 = 0). Bo Jacoby (talk) 23:33, 14 December 2008 (UTC).

In convergence you require that every ball about x contains all but finitely many points of the sequence x_n in question. What you said counts. If every ball contains infinitely many terms from the sequence, this is different from it containing infinitely many points from the sequence. So in actuality, every neighbourhood of x contains exactly one point of the sequence but still contains every term of the sequence. Maybe the following definition will help:

A point x is a cluster point of the sequence x_n if and only if given any ball, B_ε containing x, there exists an infinite subset, K, of the natural numbers such that for all k in K, x_k is in B_ε.

So the above definition implies that every ball about x contains infinitely many terms of the sequence (K is the set of all such terms for a particular ball).

Hope this helps (most importantly note the distinction between terms and points).

Topology Expert (talk) 10:23, 15 December 2008 (UTC)

By the way, did you see what I wrote about the box topology in a previous section? Topology Expert (talk) 17:28, 15 December 2008 (UTC)

I saw something you said a few days ago but I do not have time at the moment to explore things unless they are necessary for me to know in the next couple days :) StatisticsMan (talk) 03:04, 16 December 2008 (UTC)

Not the right attitude! Make sure that you explore these concepts after your exams are over. :) Topology Expert (talk) 19:32, 16 December 2008 (UTC)