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I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.

(1) Sergeyev's writing seems to contain confusion between the notions of ordinal and a cardinal numbers. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. Some of his comments here look a lot like freshman calculus errors.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) One wonders about the effectiveness of exploiting the Pirahã to "motivate" the grossone or any other form of infinity.

I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.

(1) Sergeyev's writing seems to contain confusion between the notions of ordinal and a cardinal numbers. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. Some of his comments here look a lot like freshman calculus errors.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) One wonders about the effectiveness of exploiting the Pirahã to "motivate" the grossone or any other form of infinity.

The above analysis of Sergeyev's "numerical infinity" was presented in more detailed form in this 2017 publication in Foundations of Science.

Sergeyev has since posted this feb 2018 rebuttal on the arXiv.

I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.

(1) Sergeyev does not seem to know the difference between an ordinal and a cardinal number. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. At some point one has to wonder about the thin line between freshman calculus errors and ruminations of a genius.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) The pseudoanthropology involved in the exploitation of the Pirahã to "motivate" the grossone is an echo of the pseudomathematics of the grossone itself.

I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.

(1) Sergeyev's writing seems to contain confusion between the notions of ordinal and a cardinal numbers. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. Some of his comments here look a lot like freshman calculus errors.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) One wonders about the effectiveness of exploiting the Pirahã to "motivate" the grossone or any other form of infinity.

I would like to summarize some findings concerning Sergeyev's grossone.

(1) Sergeyev does not seem to know the difference between an ordinal and a cardinal number. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. At some point one has to wonder about the thin line between freshman calculus errors and ruminations of a genius.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) The pseudoanthropology involved in the exploitation of the Pirahã to "motivate" the grossone is an echo of the pseudomathematics of the grossone itself.

I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.

(1) Sergeyev does not seem to know the difference between an ordinal and a cardinal number. These are the same in the finite case, but he postulates that his grossone has properties of both ordinals and cardinals. This certainly sounds interesting. However, he does not provide any justification for such things.

(2) Sergeyev's postulation rests at the level of a declarative pronouncement rather than a mathematical development. Perhaps his intention is to develop a new mathematical foundation, but if so he provides very little explanation. At some point one has to wonder about the thin line between freshman calculus errors and ruminations of a genius.

(3) A number of experts have emphasized at this MO discussion that whatever seems to be novel in Sergeyev's claims is subsumed under nonstandard models of arithmetic. Therefore there is nothing new here except for a refusal to provide any details, as also noted by several participants in the discussion. It is fine to popularize ideas related to infinity and infinitesimals. Where is becomes problematic is when other people's work is presented as his own insight.

(4) Sergeyev has repeatedly implied that he has superceded Cantor's set theory, and makes various claims based on Greek mathematics to the effect that the whole is greater than the part. This is also mere posturing on his part, and remains at the level of declarative pronouncement that may appeal to the masses but does not impress mathematicians who expect explanation rather than declaration.

(5) Specifically with regard to numerosities, note that all of Sergeyev's work on infinity is posterior to that of di Nasso et al. (2003). Sergeyev claims his theory is different but it is only different in that, unlike di Nasso, he does not justify any of his claims.

(6) Similar remarks apply to Sergeyev's repeated comparisons of his own "framework" with Robinson's, with the latter invariably coming out looking inferior.

(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$. If the circularity of this may bother some people that does not seem to include Sergeyev himself.

(8) Several reviewers for mathscinet have noted that the flood of publications coming from Sergeyev is repetitous and of dubious quality. See for example a recent review by Leon Harkleroad at http://www.ams.org/mathscinet-getitem?mr=3328558

(9) Sergeyev makes a distinction between "number" and "numeral". There is a lot of verbiage in his writing about this distinction and a lot seems to hinge on it. Now it is true that a real number per se is not implementable on the computer, and one needs to work with one or another representation, be it decimal numerals or some other form. That much is true and is obvious. I was not able to discern any additional meaning beyond the above in the flou artistique skillfully made to envelope this distinction in Sergeyev's writing.

(10) Sergeyev traces the history of human engagement with number systems from Pirahã to Cantor to Sergeyev in broad strokes:

"With respect to our methodology, the mathematical results obtained by Pirahã, Cantor, and those presented in this paper do not contradict to each other."

(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) The pseudoanthropology involved in the exploitation of the Pirahã to "motivate" the grossone is an echo of the pseudomathematics of the grossone itself.