Common Creationist Arguments - Morality

http://www.stardestroyer.net/Creationism/Arguments/Morality.shtml ^

[The_Reader_David]

No, I meant compact. In the case of the one point compactification, the neighborhoods of the (unique) point at infinity are the complements of compacta in the original space. In the case of compactification by ends, they are the connected components of complements of compacta which lie in the appropriate end. (Compactifying the reals by adding +infinity and -infinity is a special case of compactification by ends.)

This has already gotten more technical than most readers here can bear, so if you want more, I refer you to the Michael Spivak's Introduction to Differential Geometry, vol. 1 where compatification by ends is described in detail (in an exercise).

[The_Reader_David]

You will note that I specified set containment not cardinality in by notion of "more". You are, however, correct that my formulation was inadequate to my point. The precise statement involves looking at the definition of +infinity as an end of the real line: an equivalence class of connected components of complements of compacta--which then, together with the formal point at +infinity, form a neighborhood basis of +infinity. The set of opens in this neighborhood basis which contain 10,000 is a proper subset of the set of opens in the neighborhood basis which contain 10, and 10,000 is thus "closer" to +infinity in this topological, non-metric sense than 10.

[Kyrie]

This has already gotten more technical than most readers here can bear

Yeah, well, I think we are probably the only ones left reading, so I'm game if you are.

In the case of the one point compactification, the neighborhoods of the (unique) point at infinity are the complements of compacta in the original space. In the case of compactification by ends, they are the connected components of complements of compacta which lie in the appropriate end. (Compactifying the reals by adding +infinity and -infinity is a special case of compactification by ends.)

First question. I was studying general topology in the early 80's in the Pacific Northwest of the USA. I never heard "compacta" used as the plural of a compact set. We just said, "compact sets." Do you know if "compacta" is of recent use or perhaps regional?

Second question. Unless definitions have changed, the finite union of compact sets (compacta?) is still compact. Correct? So, for example, the union of the intervals [3,4] and [6,7] is a compact set. This compact union of disjoint intervals could be described as a "compact neighborhood of 6" that includes 4 but not 5. Then there are probably as many compact neighborhoods of infinity that contain 1000 but not 1,000,000 as there are compact neighborhoods of infinity that contain 1,000,000 but not 1000. No, I haven't determined the cardinality of either.

I know that your example was a compact interval but I would still suggest that your argument was based more on the connectedness of intervals than on compactness. Am I misunderstanding what you were trying to say?

[The_Reader_David]

Actually, I think compacta as a (singular and plural (!)) for "compact sets" is quite old, although it may be just the Bing school which uses it.

I think your query is addressed in 182.

[Doctor Stochastic]

Or take up the sword and drive the English out of France!

[Doctor Stochastic]

The natural density of a set of integers is defined to be the D(n)=(number in the set < n)/n where n is an arbitrary natural number, if this density exists as n gets arbitrarily large. Thus for any n, the number of integers divisible by three is about n/3 which gives a natural density of 1/3, likewise the even number have a natural density of 1/2 and the primes a density of (approximately) log(n)/n; the prime number density is a bit harder to show.

Comparison of sets may be done by a one-to-one correspondence. An infinite set is one which may be put into one-to-one correspondence with a proper subset of itself. For example, one can pair off each integer with its double, 1-2, 2-4, 3-6, 4-8, etc. This gives a one-to-one mapping of the integers to the even integers, showing that the set of integers is infinite. This is a different concept from that of natural density of integers.

[Doctor Stochastic]

Also ask your science teacher and guideance counselar if Truth is relative or absolute!

Truth isn't but numerical errors are.

[Doctor Stochastic]

Joseph Strauss's waltzes were embraced by Hitler and Prokifiev's music by Stalin. Does this show that waltzes and classical music are the cause of evil?

[Virginia-American]

There's been a lot of criticism of Darwinism as somehow leading to atheism, or at least a non-literal interpretation of a few verses of Scripture.

Granted it's not exactly Christian, but doesn't Darwin lead us to the conclusion that we're related to **ALL OTHER LIVING THINGS**. I personally find this awe-inspiring and highly spiritual.

Any comments?

[medved]

Yeah... Porsches and Volkswagons are pretty much related to eachother also; in fact early Porsches used a lot of VolksWagon parts and Porsche owners regularly saved money buying parts from VW dealers rather than from Porsche dealers. That does not mean that Porsches evolved from Volkswagaons via a series of lucky accidents; the Porsche was essentially re-engineered from the VW. Likewise, higher and more complex lifeforms appear to show genetic engineering and re-engineering. Evolutionists try to thrive on a salesmanship thing in which they try to insist that all evidence of change is evidence of evolutionary change and anybody who doesn't buy that is some sort of a reactionary bible-thumping fundamentalist. Don't listen to em. The biggest group of anti-evolutionists is probably mathematicians, and not fundamentalist Christians. Evolutionism amounts to an endless series of gross violations of the laws of probability; nobody with brains and talent buys it any longer.

[f.Christian]

...lizzards--liberals?

[Virginia-American]

The biggest group of anti-evolutionists is probably mathematicians, and not fundamentalist Christians. Evolutionism amounts to an endless series of gross violations of the laws of probability; nobody with brains and talent buys it any longer.

Name a few of 'em.

And, by the way, why do whales sometimes have a birth defect of legs? It seems most probable to me that it's been in their genome as long as there's been whales. Is there a non-common-descent explanation?

And that *identical* mutation that prevents us and chimps from making vitamin C?

[Kyrie]

The set of opens in this neighborhood basis which contain 10,000 is a proper subset of the set of opens in the neighborhood basis which contain 10, and 10,000 is thus "closer" to +infinity in this topological, non-metric sense than 10.

You must have posted this while I was working (with distractions) on my previous post. All right, now you are giving a privileged position to your topological basis. But your basis for that topology is not inherent in your topological space, only in your construction of it. There are other bases for the same topology on the extended line. A different basis for the topology would give a different notion of "closeness." Then your notion of closeness is not from your topology, but from your basis; i.e., from your method of compactification. Isn't that a little bit arbitrary?

[Kyrie]

Also, I take back what I said about us being the only ones left on the thread. Still, I don't think anyone else is paying attention to us...

[Kyrie]

Also ask your science teacher and guideance counselar if Truth is relative or absolute!

Truth isn't but numerical errors are.

Truth isn't relative? Or Truth isn't absolute? Not to offend, but after seeing this I wanted to know for certain which one you intended.

Or was it your intent to leave us hanging?

[Kyrie]

Your post on Cantor's diagonal argument was good. At least it worked for me. But talking real math again is so much fun I can't stop now. How about these...?

1. Cantor's theorem proves that the set of reals and the set of natural numbers, both infinite sets, have distinct cardinalities. In fact, no set has the same cardinality as its power set. So by using power sets, beginning with the set of natural numbers, we obtain an infinite ascending chain of infinite cardinals. But is there any distinct cardinal lying strictly between the (countable) cardinality of the natural numbers and the (uncountable) cardinality of the reals?
2. Is the cartesian product of nonempty sets always nonempty? (Better yet: if so, what interesting paradox does that give us?)
3. Is there any way to get mathematical notation into HTML?
4. Do you know of any more FReepers with mathematical interests?

I used to really love this stuff. The first two, anyway. I'm interested in #3 now for different reasons...

[Doctor Stochastic]

If A is correct and B is an estimate:

|A-B| is the absolute error
|A-B|/A is the relative error.

[The_Reader_David]

Actually, I think my neighborhood basis does have a privileged position: it is the maximal neighborhood basis of +infinity consisting of connected opens.

[The_Reader_David]

Your question 1 is essentially whether the continuum hypothesis is true (one can construct a 1-1 correspondence between the reals and the power-set of the natural numbers). The continuum hypothesis asserts that there are no cardinals between aleph-null (the cardinality of the natural numbers) and c (the cardinality of the reals). (Or, put another way, that aleph-one = c.) It turns out to be independent of the axioms of set theory in either the Goedel-Bernays or Zermelo-Frenkel formulations: there are models of set theory where it is true, and there are models where it fails.

For your second question, one needs to be a little more precise: that the cartesian product of a non-empty family of non-empty sets is non-empty is often considere to be almost a statement of the Axiom of Choice. In particular if such a product is always non-empty and once can choose an element out of any single non-empty set, then AC holds (the coordinates of the chosen element are a choice function for the family of sets). Again AC is independent of the other axioms of set theory, and there are perfectly fine models of set theory without AC, but the thing that goes awry with choice is not that the products are empty in the sense of being isomorphic to the empty set (the unique set with a unique set-map to any set), but that they do not have "global elements". (It turns out one can build a model of set-theory out of your favorite model of set theory with AC by considering as 'new sets' old sets with an action of the two-element group. In the new model one does not even have choice functions for non-empty families of at most two non-empty sets with at most two elements. (At most two defined in Russell's way as (for all x,y,z x=y or y=z or x=z).)

Re: 3 and 4. Not that I know of. :-(

[medved]

The biggest group of anti-evolutionists is probably mathematicians, and not fundamentalist Christians. Evolutionism amounts to an endless series of gross violations of the laws of probability; nobody with brains and talent buys it any longer.

Name a few of 'em.

Mathematicians who don't believe in evolution? I could start with Sir Fred Hoyle and Robert Bass, but the real starting point is Wistar.

The Wistar Institute Symposium was a milestone meeting held in Philadelphia in April 1966 to discuss the statistical possibility of Darwinian evolution. The conference was chaired by Sir Peter Medawar, whose work on graft rejection won him a Noble prize. By 1966, computers had progressed enough to determine statistically if random mutations alone could account for the level of evolution seen in organisms after five billion years. After a heated debate and several meetings, the Wistar Symposium deemed this statistically impossible.

Furthermore, many of the scientists at Wistar came forward to state that the fossil record did not support evolution. Few fossils showing transitional stages between species had been found. Arguments also came up about advanced organs such as the eye and that 5 billion years was not enough time for these organs to evolve.

For more details about the Wistar Institute Symposium, see the following links:

• Wistar Destroys Evolution
• The Anti-Darwinian Scientists

Wendell Bird, author of the article at the second site, is the author of a two-volumn destruction of modern evolutionary theory published by the Philosophical Society of New York, which could hardly be categorized as a bunch of bible-thumping rednecks or fundamentalists.