A Set Theory for Scientists and Engineers

Engineers know that they can land a man on the moon without using the Lebesgue integral and they will never encounter Skolem paradox in their nuclear reactor design. Intuitive Set Theory (IST) defined here, de-emphasizes concepts that are not required by scientists in their practical work.

AXIOM OF COMBINATORIAL SETS: A set as important as the powerset of Cantor is what I call the combinatorial set of \aleph_0, which is defined as the set of all subsets of \aleph_0 with cardinality \aleph_0. Axiom of Combinatorial Sets (ACS) says that \aleph_1 is equal to the combinatorial set of \aleph_0. Even though, the combinatorial set is a subset of the powerset, it can be shown that powerset and combinatorial set have the same cardinality.

AXIOM OF iNFINITESIMALS: First of all, let us note that corresponding to every real recursive number it is possible to visualize an infinitesimal attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite binary sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. From this, we can say that an infinitesimal is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. The Axiom of Infinitesimals (AI) says that the unit interval is a set, with cardinality \aleph_0, of infinitesimals. We call an infinitesimal an relement and the elements in it figments, claiming that not even the axiom of choice can pick a figment from an relement.

INTUITIVE SET THEORY: We define IST as the theory we get when AI and ACS are added to ZF theory. The discerning reader will easily recognize that the notion of a figment will not allow nonLebesgue measurable sets in IST. Also, the fact that \aleph_0 is the cardinality of the set of infinitesimals in a unit interval, provides us with a way to circumvent the Skolem paradox.

IN A NUTSHELL: If only relements are allowed in set theory, it is enough for scientists for all practical purposes. If all elements of ZF theory are allowed, then set theorists can live happily in "Cantor's heaven".

Category:

Education

Tags:

• math
• set
• theory
• intuitive
• continuum
• hypothesis
• infinitesimal
• ordinal
• cardinal
• transfinite
• cantor

[Muffinman10123]

Also, the combinatorial set axiom seems to follow logically from the axiom of power set...

[kautilya33]

Cardinalities of the combinatorial and powerset are the same, but the combinatorial axiom does not follow from the powerset axiom. For more details, type "intuitive set theory" without quotes in google search box.

[Muffinman10123]

Hmm. I thought that bonded sacks, which have figments as its elements, are defined by the fact that they cannot have elements picked by the axiom of choice. I am new to set theory and all of these concepts in general, sorry..

[kautilya33]

IST merely defines and introduces the notion of a bonded sack with untouchable figments in it, and in no way interferes with the elements of ZF theory and the axiom of choice.

[Muffinman10123]

This omits the Axiom of Choice?

[kautilya33]

Axiom of Choice is very much part of IST. It can choose any element from a set containing infinitesimals and figments.

[tommyrjensen]

There is already a set aleph_1 used in ZF, it is defined as the set of all countable ordinals. It is clearly not the same as the combinatorial set of aleph_0, since every finite ordinal is an element of aleph_1 but not of the combinatorial set of aleph_0. So what does your ACS really mean? It looks like you want to use the same name for two altogether different sets.

[tommyrjensen]

It is easy to prove in ZF that the unit interval is an uncountable set. Thus if you add AI to ZF you apparently get an inconsistent theory. It does not seem of much value then.

[Gargonzolo]

@Dzikslol thumbs down for MIDI

[Dzikslol]

Thumbs up for Bach

[ahhhhjjjj]

@CHistrue You sounded like the Nietzsche of Mathematics there for a second. Hahaha. Just teasing :P

[CHistrue]

In my video's (please view), I presented an elementary philosophical argument as to why I believe that the Continuum Hypothesis is correct. My main axiom is the notion that levels of Infinity higher than Aleph Nought are not bound by discreet binary reality, and thus that they cannot be multiplied by zero in any real sense. Calculus=limits, but "thing-in-itself" transcends limits. Cantor was correct in his insights, I believe, making Calculus a reality on one level transcended on another.