Descriptive definitions of potential and actual infinities

Zhu, Wei; Lin, Yi; Gong, Na; Du, Gonghuan

doi:10.1108/03684920810863372

Cited by 17 publications

(12 citation statements)

References 8 publications

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“…First, from Cantor to Zermelo, the principle of one‐to‐one correspondence appeared everywhere. Just as what is pointed out in Example 4 (Zhu et al , 2008) that when applied to infinite sets, the principle of one‐to‐one correspondence is also a procedure of listing, and before the procedure is exhausted, the process is forever a present progressive tense (going). Therefore, what is facing the procedure must be a potential infinity.…”

Section: Compatibility Of Both Kinds Of Infinities In Modern System Of Mathematicsmentioning

confidence: 95%

“…In Zhu et al (2008), we have mainly clarified differences and connections between potential and actual infinities. The core content of our work there contains two aspects:each actual infinity has to be a present progressive tense (going), which is transited to a perfect tense (gone); andeach potential infinity must be a forever present progressive tense (going) strengthened from a present progressive tense (going).Then, from two different angles, one is “eventually reach” or “never reach,” and the other “procedure of listing” or “exhausted procedure of listing,” we provided two concrete models or specific realizations for present progressive tense and perfect tense.…”

Section: Some Elementary Conclusionmentioning

confidence: 99%

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Wide‐range co‐existence of potential and actual infinities in modern mathematics

et al. 2008

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PurposeThe paper aims to introduce the concepts of potential and actual infinities.Design/methodology/approachA conceptual approach is taken.FindingsIt is a common belief that both Cantor and Zermelo completely employed the thinking logic of actual infinities in the naive and modern axiomatic set theory, and that Cauchy and Weierstrass completely applied that of potential infinities in the theory of limits. However, when it explores in depth the essential intensions of both potential and actual infinities, and after sufficiently understanding the difference and connections between the infinities and revisiting the realistic situations on how the concept of infinities has been employed in modern system of mathematics, it is discovered that in set theory, the thinking logic of actual infinities has not been applied consistently throughout, and that in the theory of limits, the idea of potential infinities has not been utilized consistently throughout, either. As for those subsystems involving the concepts of infinities of modern mathematics, they generally contain both kinds of infinities at the same time. As a matter of fact in modern mathematics and its theoretical foundation, one only needs to slightly analyze and dig deeper, one will see the reality that the thinking logics and method of analysis of employing both kinds of infinities are everywhere implicitly.Originality/valueThe authors show the first time in history that the system of modern mathematics is not consistent as what has been believed.

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Section: Compatibility Of Both Kinds Of Infinities In Modern System Of Mathematicsmentioning

confidence: 95%

Section: Some Elementary Conclusionmentioning

confidence: 99%

Wide‐range co‐existence of potential and actual infinities in modern mathematics

et al. 2008

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“…At the end, we should point out that the thinking logic and the reasoning method of coinciding poi with aci are inherently part of modern mathematics (Zhu et al, 2008a), and that our discussion in the previous section tells us the fact that the thought convention of poi being not the same as aci is also an inherent part of modern mathematics. On the other hand, these results mean that the thought and method of equating poi and aci and the thought convention of poi -aci are not artificially added into the system of modern mathematics and its theoretical foundation.…”

Section: Some Plain Explanationsmentioning

confidence: 98%

“…And, consequently, "for each x, a certain conclusion holds true" now is equivalent to "for all x that certain conclusion holds true." However, at this junction, we needs to point out: when one says to select an arbitrary entity a out of an infinite universe of discourse, or he talks about each entity from the universe, as we have discussed in Zhu et al (2008a), he is talking about a procedure of listing. And, any non-terminating procedure of listing, before it is finished, is forever a present progressive tense (going).…”

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confidence: 98%

Modern system of mathematics and a pair of hidden contradictions in its foundation

et al. 2008

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PurposeThe paper's aim is to show a pair of deeply hidden contradictions in the system of mathematics.Design/methodology/approachThe paper takes a conceptual approach to the problem.FindingsIt is indicated that it is an intrinsic attribute of modern mathematics and its theoretical foundation to mix up the intensions and methods of two different thoughts of infinities, which provides the basis of legality for using the methods of analysis, produced by combining the two kinds of infinities, in the study of the modern mathematical system. In this paper, by exactly employing the method of analysis of mixing up potential and actual infinities, we card the logical and non‐logical axiomatic systems for modern mathematics. The outcome of our carding implies that in modern mathematics and its theoretical foundation, some axioms implicitly assume the convention that each potential infinity equals an actual infinity, while some other axioms implicitly apply the belief that “each potential infinity is different of any actual infinity.”Originality/valueBy using the concepts of potential and actual infinities, the authors uncover two contradictory thinking logics widely employed in the study of mathematics.

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“…Since, we now know the descriptive definitions of actual and poi, for more details on these definitions, please consult with Zhu et al (2008i), all proven above by using mathematical induction is that for any natural number j , a term a k j of { a i } i =1 ∞ can be picked to satisfy a set of desirable conditions. So, the non‐terminating process of getting one more term a k j out of { a i } i =1 ∞ can be carried out indefinitely.…”

Section: Proposition 31mentioning

confidence: 99%

Systemic yoyo structure in human thoughts and the fourth crisis in mathematics

et al. 2008

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PurposeThe paper aims to show the existence of the systemic yoyo structure in human thoughts so that the human way of thinking is proven to have the same structure as that of the material world.Design/methodology/approachParallel comparison is used to reveal the underlying structure existing in human thoughts.FindingsAfter highlighting all the relevant ideas and concepts, which are behind each and every crisis in the foundations of mathematics, it becomes clear that some difficulties in the authors' understanding of nature are originated from confusing actual infinities with potential infinities, and vice versa. By pointing out the similarities and differences between these two kinds of infinities, then some hidden contradictions existing in the system of modern mathematics are handily picked out. Then, theoretically, using the authors' yoyo model, it is predicted that the fourth crisis in the foundations of mathematics has appeared. And, a plan of resolution of this new crisis is provided.Originality/valueThis paper shows the first time in history that human thought, the material world, and each economic entity, share a common structure – the systemic yoyo structure. And it proves the arrival of the fourth crisis in mathematics by using systems modeling and listing several; contradictions hidden deeply in the foundations of mathematics.

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Descriptive definitions of potential and actual infinities

Cited by 17 publications

References 8 publications

Wide‐range co‐existence of potential and actual infinities in modern mathematics

Wide‐range co‐existence of potential and actual infinities in modern mathematics

Modern system of mathematics and a pair of hidden contradictions in its foundation

Systemic yoyo structure in human thoughts and the fourth crisis in mathematics

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