It is extremely a system regarding anything whose common affairs are governed by basics create and also for and therefore all of the propositions, and just the individuals, was correct that’s based on the fresh new rules because of the a great limited number of logical process
But above all I would like to employ another because most critical one of the numerous issues and is expected having reference to new basics: To show they are perhaps not contradictory, which is, one to a definite quantity of analytical tips reliant him or her can be never ever lead to inconsistent efficiency.
Within the geometry, the newest proof of brand new being compatible of your rules would be effected because of the building an appropriate field of numbers, in a way that analogous connections amongst the amounts of it profession correspond on geometrical basics. People contradiction in the deductions in the geometrical rules need certainly to with this end up being identifiable on arithmetic on the field of amounts. In this way the required facts on the being compatible of the geometrical principles was designed to rely on the fresh theorem of compatibility of the arithmetical maxims.
The brand new axioms out of arithmetic is actually fundamentally very little else compared to recognized regulations off calculation, adding the fresh axiom away from continuity. I recently built-up him or her and also in so doing replaced the fresh axiom off continuity of the a couple simpler concepts, particularly, the new really-recognized axiom out of Archimedes, and you may a different axiom essentially the following: one to quantity setting a network away from things which is capable of no longer extension, for as long as all other maxims keep (axiom off completeness). I’m convinced that it needs to be you’ll be able to to acquire an effective lead evidence on the being compatible of your arithmetical concepts, as a mindful investigation and you can appropriate amendment of your own recognized ways of reasoning from the idea regarding unreasonable wide variety.
Indeed, if research on the being compatible of concepts is fully done, the newest second thoughts that happen to be expressed sometimes to what existence of one’s over program regarding genuine number becomes completely groundless
To show the necessity of the problem out of getiton several other section off glance at, We are the following observation: If inconsistent characteristics feel allotted to a concept, We state, you to mathematically the idea doesn’t exists. Thus, like, a bona-fide amount whose square are -l will not are present mathematically. But if it can be proved the features allotted to the concept will never end in a contradiction by software out of a small amount of logical techniques, I point out that new statistical lifestyle of style (such, off a variety or a features and therefore touches certain requirements) are and therefore turned-out. In the case in advance of you, where our company is concerned with the fresh basics out of real numbers during the arithmetic, new proof of the newest compatibility of maxims is at the fresh same day the brand new evidence of new statistical life of your over program away from actual wide variety or of the continuum. New entirety of genuine quantity, i. e., the newest continuum depending on the perspective merely indicated, isn’t the entirety of all of the possible collection within the quantitative fractions, otherwise of all it is possible to legislation considering that areas of an elementary succession get go ahead. I believe, the idea of this new continuum is precisely logically tenable within sense just. It looks to me, in fact, that this corresponds greatest and exactly what experience and you may instinct give united states. The thought of new continuum otherwise that of the computer of all the services can be obtained, after that, for the the same experience as the program regarding integrated, mental number, like, otherwise once the Cantor’s high groups out-of amounts and you can cardinal number. Having I’m believing that the presence of the latter, just as regarding this new continuum, is turned out in the same way I’ve described; instead of the computer of all the cardinal numbers or of all Cantor s alephs, where, since the tends to be revealed, a system away from basics, appropriate during my sense, can not be created. Possibly ones solutions are, ergo, based on my terminology, mathematically non-existent.