Introduction to axioms

Before you start reading further read this:

http://en.wikipedia.org/wiki/Paradox

http://en.wikipedia.org/wiki/Novikov_self-consistency_principle

http://en.wikipedia.org/wiki/Closed_timelike_curve

Now, I’m assuming you had read the articles (links above). In order to have a fallacy, you need to bump into a contradiction in given sense. In order to be able to do so, you need to compare the actual situation with the (always) true statement. In order to have the latter, you need to start with defining the axioms, ie. consistent constraints of the logic behind your world. One.

The paradoxes given in the articles below are not even paradoxes in the proposed sense, as they lack rigorous approach to logic. Still, a couple of things are worth discussing here.

Firstly, falsidical paradoxes, ie. paradoxes where your implications are not true due to the fact you made a step that is impossible in the defined world. Beware every step, ie. pay attention to details. Besides this, in one world this step may lead to contradiction; in another one, not; potentially, as claimed by Gauss, this may have eventually an objective answer. Due to the lack of objective constraints (this will be addressed later), we cannot now talk about truth in objective sense.

Secondly, nothingness, ie. the idea close to mathematics. We assume that there exists an empty set. The contrary: can we create a number p_1p_2..p_n from infinite number of prime numbers? If yes, such \mathbb{P} would be one we would not be able to show as sum of two primes, thus contradicting the Goldbach Conjecture, hence I tend to think that such construction does not exist, thus the inquiry. Next, infinity- how to judge which path is right? Cantor’s, Kronecker’s?

In order to go deeper, if we assume only natural numbers exist, thus \sqrt{2} would have to be shown as “another dimension” value (with respect to 2), maybe even using what I will be describing later in the BTW section. Then, we’d potentially have prims the most important numbers, creating the place for geometrical understanding of numbers.

Now, for a couple of axioms, take http://en.wikipedia.org/wiki/Tarski’s_axioms

http://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theory

Now, take a look at

http://en.wikipedia.org/wiki/Russell%27s_paradox

Therefore, finally look at:

http://en.wikipedia.org/wiki/Zermelo_set_theory

http://en.wikipedia.org/wiki/General_set_theory

http://en.wikipedia.org/wiki/Kripke-Platek_set_theory

And, finally, ask yourself simple questions, what is a set? what happens if you choose a couple of objects (to the rest of the objects)? how many objects do we have in an infinite set? If, infinite amount, then can we create such set? (see BTW section for fruitful analysis).

This is very short, way too cursory, so expect a more accurate example on a given subject in the area. We will try to go deep with a couple of axioms in this blog. This will take time.

An example regarding the use of axiom exploitation (yet, incomplete and nonobjective), when taking

We shall embody this viewpoint in a principle of self-consistency, which states that the only solutions to the laws of physics that can occur locally in the real Universe are those which are globally self-consistent. This principle allows one to build a local solution to the equations of physics only if that local solution can be extended to a part of a (not necessarily unique) global solution, which is well defined throughout the nonsingular regions of the spacetime.

(on Novikov, Wikipedia) into consideration, we’d have time loop logic based on prime-based channels, defined further using our BTW section.

BTW, It is known that lim_{n\rightarrow\infty}(p_1p_2...p_n)^{{{1}\over{p_n}}}=e. By definition, we have e=lim_{n\rightarrow\infty}(1+{{1}\over{n}})^n=e,ie. e=lim_{n\rightarrow\infty}({{n+1}\over{n}})^n=e, thus we have n+1=n(p_1p_2..p_n)^{{{1}\over{np_n}}}, speaking in terms of limits, which clearly resembles cyclotomy, hence its connection to cyclotomic factorization, ie. z^p-y^p=(z-y)(z-ay)(z-a^2y)..(z-a^{p-1}y), where a is de Moivre’s number. This inexplicably astonishing and vulnerable connection between multiplication and addition, born in divine admiration, shows clear interest in cyclotomic fields.

Saying that aloud, transition from N(n)\rightarrow N(n+1) has much to do with multiplication by primes. As there are exponents involved, it’s most likely about creating multidimensional “capacities”, which I had mentioned years ago.

A couple of quotations:

The time in which I write … has a horribly swollen belly, it carries in its womb a national catastrophe … Even an ignominious issue remains something other and more normal than the judgment that now hangs over us, such as once fell on Sodom and Gomorrah … That it approaches, that it long since became inevitable: of that I cannot believe anybody still cherishes the smallest doubt. … That it remains shrouded in silence is uncanny enough. It is already uncanny when among a great host of the blind some few who have the use of their eyes must live with sealed lips. But it becomes sheer horror, so it seems to me, when everybody knows and everybody is bound to silence, while we read the truth from each other in eyes that stare or else shun a meeting.

Germany … today, clung round by demons, a hand over one eye, with the other staring into horrors, down she flings from despair to despair. When will she reach the bottom of the abyss? When, out of uttermost hopelessness — a miracle beyond the power of belief — will the light of hope dawn? A lonely man folds his hands and speaks: “God be merciful to thy poor soul, my friend, my Fatherland!”

— Thomas Mann, Dr. Faustus (1947, written in 1945)

“The purpose of life is a life of purpose.”
— Robert Byrne

Our prime purpose in this life is to help others. And if you can’t help them, at least don’t hurt them. The Dalai Lama

“The ideals which have lighted my way, and time after time have given me new courage to face life cheerfully, have been Kindness, Beauty, and Truth…”, Albert Einstein

“Hide not your talents, they for use were made.” Benjamin Franklin

Do not squander time for that is the stuff life is made of.
Benjamin Franklin

“Pursue some path, however narrow and crooked, in which you can walk with love and reverence.”
Henry Dawid Thoreau

BTW2, From Larry Freeman’s blog Fermat’s Last Theorem, a sample example of use of infinite descent combined with a trivial 20-step proof. Though, it might be nice to give a couple of quotations from wise men as well as things we may all learn from. If not from a simple proof itself, then from ideas behind the argumentation etc.

Theorem: Relatively Prime Divisors of an n-power are themselves n-powers.
This theorem says that if gcd(v,w) = 1 and vw = zn
Then, there exists x,y such that v = xn, w = yn

(1) So, we start with gcd(v,w) = 1, vw = zn
(2) Assume that v is not equal to any number xn
(3) v ≠ 1 since 1 is an xn power
(4) Now, v is divisible by a prime number p. [Fundamental Theorem of Arithmetic]
(5) So, there exists k such that v = pk
(6) p divides z since zn = vw = pkw [By applying Euclid’s Lemma]
(7) So, there exists m such that z=pm
(8) So, zn = vw = pkw = (pm)n = pnmn
(9) Dividing p from both sides gives us:
kw = p(n-1)mn
(10) From Euclid’s Lemma, p divides k or w.
(11) It can’t divide w since it already divides v and gcd(v,w)=1. Therefore, it divides k
(12) We can apply this same argument for each p in p(n-1)
(13) So, we can conclude that p(n-1) divides k.
(14) So, there exists V such that k = p(n-1)*V
(15) So, kw = p(n-1)mn = p(n-1)*V*w
(16) Dividing p(n-1) from both sides gives us:
Vw = mn
(17) Now, gcd(V,w)=1 since V is a divisor of v and gcd(v,w) = 1
(18) Likewise, V cannot be an n-power. If it were, then v = pnV would make v an n-power which goes against our assumption.
(19) Finally, V is less than v since p(n-1) > 1.
(20) Thus, we have a contradiction by infinite descent.

This poem, very different from the volume of peotry I will be releasing soon, is dedicated to one person who does not know about it.

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About misha

Imagine a story that one can't believe. Hi. Life changes here. Small things only.
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