What is transfinite induction? What is a Turing jump operator? Degrees of unsolvabilty?

To recapitulate from where we left off yesterday, can one write a number so large so as to the entire concept of writing that number becomes meaningless?

But trifles aside, we will investigate the philosophical problems of induction. Do note: a paper stating if RH can be proven by induction does exist.

Our approach would be different. Before we tackle transfinite one we have to understand mathematical induction itself. Immediately the fact that jumps up is that why should induction work on numbers of arbitrarily infinite length? Suppose our entire history of the universe were binary coded and a tape started reading it. It has infinite length at each ends and the tape when reaches the time universe ends, we assume, that the time starts traveling background. So as the number in the meter of an Unbreakable wrist-watch shows that billions of years elapsed, and then a la Asimov/Hawking, time starts traveling back while the wrist-watch numbers keep increasing. Isn’t this an example of paradoxical event from…say, “overflow”.

Induction is related to entailment. Now one can easily devote tomes on the latter, but I will put the shortest disproof of entailment:

Consider *modus ponens*

if p entails q

if p

then q

or this can be rewritten as ( ((p → q) → (p))→q).

Now bear with me. Now all of mathematics can be rewritten in formal language of symbolic logic, where entailment glues and litters the proofs like broken shells and weeds in a sandy beach.

Now I have another problem.

Take MP. Take an invalid MP. Now send them as message to outer space as sample of logical thought. Suppose they receive the invalid MP first:

if p entails qif p

then p

It is an invalid form of MP. So they receive the second batch and unpack to see

if p entails qif p

then q

which is of course the valid form of MP.

However, the aliens will be terribly perplexed! So they had the wrong one first and they put a chip in their brain programmed with that form to evaluate life differently.

So every time a match strikes and fire is seen, they see the correct form of MP.

Then they are given the wrongful form of MP in their brain.

So every time a match strikes and fire is seen. But if a match is struck, then match is struck. That’s it. No fire. Nothing happens.

Now we come to the fun part. Suppose there are two envelopes. One has the correct form another the incorrect form. Tell the alien to randomly choose one. Be advised the chip in his brain is blank. So if he picks the invalid one, then he programs the chip the schema”

if p then qif p

then p

Now to confuse him we tell him to open the second one. He is still wearing the “wrong chip”, whose program is if p then q, p, then p. Thus, when he reads the correct one:

if p then q

p

q

Well, but the way his mind works is different. He is programmed with the invalid form ( ((p → q) → (p))→p)

So let’s run through this. Basically the form is if p then p, if p, well then p. Now reading second envelope, he reads if p then q, then if p it could be either p or q. It could be p because that is the wrong program in his chip.

My question is who is to say the first MP is valid than the second one. These are all arbitrary MAN MADE symbolic manipulations and not handed down in Stone Tablet but taught at schools and colleges with papal authority as if it is “IT”. Ultimate word of God embedded in Bible of logic which everyone has to take by belief.

If we lived in an alien planet where a man could penetrate wall and bring out stuffs from other side, then the logical structure of that reality would be very different. Or if Jesus did his miracles, again the mathematics would be 1=1+1+1+1…… if you want to model how he pulled all loaves of bread from a single container.

But math IS a religion. It is magic. It is founded on beliefs and axioms which are peer-reviewed. In class I read Pojman’s (West Point grad) book and I can relate to what he said that ultimately we may come to a realization where everything in this life is maya or facade (my description). We have been lied to. Senses lied to us.

For more of my motivation and where I am coming from click here.