review of mistakes occurring most often #130; 2nd ed; Euclid's Infinitude
- From: "plutonium.archimedes@xxxxxxxxx" <plutonium.archimedes@xxxxxxxxx>
- Date: Sun, 20 Sep 2009 00:51:08 EDT
Alright, let us review the mistakes that occur most often in doing a proof
of the Euclid Infinitude of Primes.
(1) The first mistake most make is they only do one of either the
direct or indirect or a mix up of the two. Some may speak out
whether they are doing the Indirect or Direct, but why not just
provide both? So as to help the reader and help the author to
insure no mixup in methods.
(2) The Direct method, if the author is focused on increasing
set cardinality has a good chance of being error free. So the
Direct seems to be easier and less error prone. If there is a
usual mistake in the Direct is that they often forget to have
the definition as the first step.
(3) The Indirect is the one that causes the most trouble. And the
first mistake that most make is that they start with the Hypothetical
assumption of the Reductio ad Absurdum. They should start with
the definition as first step.
(4) If they do not miss the definition in the first step, then the big
error occurs usually after they form W+1, for they usually cannot
see that in this Hypothetical structure that W+1 is necessarily prime.
They mistakenly look to examples such as 1+2x3x5x7x11x13 = 59x509
forgetting that they are in hypothetical space where that number is
necessarily prime since all the primes that exist were used. This is the
most often and biggest mistake in the Indirect. But recently I see there
is another big mistake frequently made, calling it the disconnected-
contradiction.
(5) Disconnected Contradiction, or you can call it something else, and
is where a person finds a contradiction in the Indirect but places that
contradiction with incorrect inferences. In Euclid's IP, after forming
W+1, several people use lemmas or theorems about prime divisors
and fetch a contradiction. But they mistakenly think that the contradiction
is to the reductio ad absurdum paragraph, but rather it is to the
subparagraph of whether W+1 is prime or composite. Their mistake is
that, in order to release the reductio ad absurdum, they have to find
a new prime larger than the largest prime in the hypothetical assumption.
So this is an error that comes up frequently, where the author finds a
contradiction, but it is only to a subparagraph, not the entire argument.
Now I wish I could use the Fitch Symbolic Logic format of vertical lines
and horizontal lines to keep track of steps and inferences, but that is
difficult on usenet posts, so I have to settle with numbering such as
(3) and then for a subparagraph (3.0.1). So that in the Indirect Method
the numbering can highlight the fact that a subargument was made
inside a larger argument and that the contradiction reached was for the
subargument and not the larger argument.
(6) There is this mistaken viewpoint out there, held by many people that
a Indirect Method is a hard drive pushiness to achieve a contradiction, any
old contradiction and then you end the proof. But that is a silly notion, since proofs and definitions and concepts in math are so tightly wound up in precision that a person should be able to realize what the contradiction
has to be in future steps to release the hypothetical assumption. In the
case of IP, that contradiction is to produce a new larger prime for the list
of finite primes. So the silly notion that seems widespread in mathematics
is that a proof is a listing of a bunch of statements and that the end of the proof is a salad-bar-arrangement where you can select what contradiction
you desire. If that view were true, then math would be too loose and hole
ridden to not be the science of precision and where Reductio ad Absurdum
could not even work. The reason it works is that there is only one contradiction that can release a given hypothetical assumption.
(7) There is also this hideous notion about Euclid's IP that there are tens,
hundreds of independent and valid proofs using W+1. Well, math is so
tightly winded up in precision, that there is only one valid structure for
Euclid IP either Direct or Indirect. And once you know the structure--
define prime
hypothetical assumption
P_k is the largest and last prime
form W+1
W+1 necessarily prime
contradiction
Primes infinite
In that above structure there is only one valid channel. You can add on
extra irrelevant garbage and still be valid. But if you omit pieces of the
essential structure, you have an invalid attempt. Now there is an independent Topology proof of the Infinitude of Primes, but all proofs
that use W+1; it has only one valid channel.
I am sure there are more mistakes that frequently come up but the above is a good starting point.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.
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