Articles
Abstract
I Empirical Fact: 0 is not a Number but represents “nothing”
Oct. 2022. In science today, 0 is undisputedly considered to be a number. But 0 marks an empty place in the sequence of digits of place value numbers. This empirical fact necessarily presupposes the meaning "nothing" of numbers, i.e. "no number". This statement was valid for three thousand years. In the 16th century, due to philosophical and religious reservations about the metaphysical Nothing, it was contradictorily reversed into its opposite, "something" and "number", thereby violating the tertium non datur. This wrong rationalistic definition is stipulated axiomatically. However, it shows no effects on applied mathematics, but causes contradictions in set theory. The axiomatic re-implementation of 0 to represent “nothing” initiates a re-evolution of the foundations of mathematics.
To the comments I Empirical Fact: 0 is not a Number but represents “nothing”
II Inconsistence of the transfinite number ω and the set ℕ
April 2023. For more than 2 millennia, Aristotle's "infinitum actu non datur", "there is no limited infinity", dominated the perception of the infinite. End of the 19th century Georg Cantor introduced the actual infinity or transfinite with the postulate of limits, even levels in infinity, with his set theory. The result was a foundational crisis of mathematics. Cantor's doctrine was not axiomatically founded. The axiomatic system of Zermelo and Fraenkel then formally legitimized the transfinite. The crisis is considered to be overcome thereby. Cantor's transfinite was initially highly controversial, but ultimately prevailed. Today, in its axiomatic form, it constitutes one of the foundations of mathematics. In the following it will be shown that both, the axiomatic and Cantor's justification of the transfinite number ω and the set ℕ
To the comments II Inconsistence of the transfinite number ω and the set ℕ
III Cantor's faults in his doctrine of the transfinite
April 2023 Existence of the transfinite is disproved by article II. Hypothetically presuming existence, Cantor's works is scrutinised. The antagonistic disparity of transfinite ordinal and cardinal numbers is rebutted, the continuum hypothesis is demonstrated to be wrong. The ostensible equality of sets of points of extremely different geometrical objects, such as a small segment and n- dimensional space, is falsified. The paradox of equivalent transfinite sets and subsets, seemingly suspending Euclid's 5. axiom, "the whole is greater than the part", is resolved. Euclid's axiom is safeguarded.
To the comments III Cantor's faults in his doctrine of the transfinite
IV Inconsistency of the empty and the transfinite Set
April 2022. The empirical proof is provided that the axiom that requires the empty set contradicts reality. The contradiction with nature also applies to Cantor's idea of the transfinite, as Hilbert has acknowledged. Accordingly, empty and transfinite sets are based on rationalistic theories that contradict empiricism and are therefore necessarily wrong. However, Hilbert did not draw this conclusion but demanded the mathematical consistency of the transfinite. The formal proof that the transfinite set does not exist requires the refutation of the ZFC-axiom of infinity. This axiom presupposes the empty set, that according to empirical evidence does not exist. The axiom of infinity is therefore contradictory. The empty set does not appear in Cantor's doctrine, ZFC contradicts Cantor.
To the comments IV Inconsistency of the empty and the transfinite Set
V Comprehensive meaning of ∞ and elimination of contradictions
Nov. 2022. The symbol ∞ can only be completely understood if a standard length is assumed. The axiom of infinity defines ∞ by the potentially infinite sequence of standard units of length on the axis of Euclidean space. When non-standard quantities are considered, counts are greater or less than ∞. This also applies to natural numbers resulting from mapping line segments. Cantor premised a single transfinite set of natural numbers, in fact many unlimited infinite sequences of these numbers exist. In contrast to previous theory, consistent calculation rules apply, e.g. ∞ + n > ∞ and ∞ + ∞ = 2 ∞.
To the comments V Comprehensive meaning of ∞ and elimination of contradictions
VI Planck units refute converging infinite sequences and limits
Only finite converging sequences can be justified
Nov. 2022. The irrational numbers are defined by limits of potentially infinite converging sequences of rational numbers that require infinite sequences of digits of the irrational numbers. Real numbers can be generated by mapping of segments. However, the subdivision of distances is limited by the Planck length, which only allows finite digits when mapped to real numbers. Potentially infinite sequences of ever smaller differences of distances and numbers as well as limits can no longer be justified. They are replaced by finite sequences with limitation of digits. This also applies to analysis, the differential and integral calculus, the Planck-length causes limitation. Limits, Δ x → 0, n → ∞ and infinitesimals do not occur. Limitation and Δx = 0, "nothing", are the decisive criteria. In practice already always limitation-values are determined, the calculation must be terminated sometime.
VII The disproof of undecidability of Gödel's proposition
Dec 2022. Kurt Gödel constructed a proposition of the theory of natural numbers in 1931 for which neither a proof nor a refutation should exist, though it is true. The true meaning of 0, "nothing", discloses new prospects of proof. Propositions about "non- existence" can be proved by equivalent propositions about "nothing". Gödel's proposition, based on the "non-existence" of its proof, is proven and decided by "nothing" of proof. The theory of the natural numbers is complete, the axiomatic system that Gödel premised, is incomplete. It is completed by the axiom of "nothing" of proof.
To the comments VII The disproof of undecidability of Gödel's proposition
VIII Fiction of the insignificance of the syntax of mathematical logic
Dec 2022. David Hilbert saw a “completely satisfying way” to escape the antinomies of set theory. Through his formalism he calls for the insignificance of the syntax of mathematical logic. Semantics should then only allow consistent interpretations. The overview of the syntactic signs, formulas and rules shows that the interpretation is already inherent. The alleged insignificance is an unreal fiction, which also did not protect against inconsistency.
To the comments VIII Fiction of the insignificance of the syntax of mathematical logic
IX Nothing, “Nothing”, nothing, “nothing”
Dec 2022. The confusion surrounding the meaning of 'nothing' by upper and lower case, with or without quotation marks is resolved in the article on hand. The definition of 0 as "number" ² and Ø as "empty set" ³ , contrary to the true meaning "nothing", implied the wrong axiomatic postulate of transfinite numbers and sets. The ban on "nothing" that began in the 16th century also deprived mathematics of a crucial potential of proof. Equivalent propositions about "nothing" and "non- existence" prove each other. The "non-existence" of a proof is proved by the existence of "nothing" of proof theory. This statement leads to the proof of Gödel's supposedly unprovable theorem and the exceedance of the incompleteness theorems. Δx = 0, “nothing”, is the decisive criterion of a revised analysis, i.e. the differential and integral calculus. The reimplementation of “nothing” initiates a re-evolution of the foundations of mathematics. Philosophical and mathematical Platonism are refuted by Nothing”, “Nothing”, nothing and “nothing”.
X Resolving the antinomies
Firstly, the difference between antinomies and paradoxes, which is not clearly defined in the literature, is defined. Antinomies are contradictions that cannot be (or appear to be) resolved. The paradox is only apparently contradictory and can be explained logically.
In fact, there are no antinomies in nature, only contradictory attempts by man to explain them. The reason for the apparent contradiction can be uncovered and eliminated, only paradoxes exist. Antinomies are apparent logical errors of thought. In the following, only the 'antinomies' that have played or still play a decisive role in mathematics will be discussed.
XI Crisis of thought of the theorists of the foundations of mathematics
April 2022. A supposedly consistent theory of the transfinite is postulated as part of the foundations of mathematics, despite being confirmed already a century ago to be inconsistent with reality. This thinking of the theorists of foundations of mathematics reveals disregard for the criteria of science. A theory that denies realness is necessarily and demonstrably wrong. The empty set, that ist postulated to justify the transfinite, fails in reality. The insistence on the infinite subdivision in contradiction to Planck, the fiction of insignificance of the syntax of mathematical logic, the failure to resolve the antinomies and the refusal to perceive the disproof of Gödel's incompleteness theorems demonstrate a disruption of thinking. The foundational crisis of mathematics, that evolved before the 20th century for decades, was not overcome by the measures taken to overcome it, but exacerbated.
To the comments XI Crisis of thought of the theorists of the foundations of mathematics
XII Crisis and reEvolution of the foundations of mathematics
The antinomies and paradoxical properties of Georg Cantors set theory led to a crisis of the foundations of mathematics at the end of the 19th century. The axiomatic system of Zermelo and Fraenkel, ZFC, was intended to create foundations free of contradictions. The vast majority of mathematicians and logicians assumed that this had been achieved. In the book „Nichts“ it is shown that crucial foundations are based on metaphysical assumptions that contradict reality and are therefore false.
The crisis was not overcome by ZFC, but deepened. The author also counts Gödels demand of incompleteness of the theory of natural numbers and the claim that the syntax of mathematical logic is meaningless among the symptoms of the crisis.
However, the phenomenal development of applied and pure mathematics is not affected by this. But an adequate evolution of the foundations has not taken place, and the banishment of "nothing" from mathematics and logic is closely related to this.
The errors of the current foundations and their correction are shown in the following summary of Articles I - XI:
To the comments XII Crisis and reEvolution of the foundations of mathematics