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1. • CommentRowNumber1.
   • CommentAuthorHermann Sommer
   • CommentTimeJan 12th 2025
   • PermaLink
   Author: Hermann Sommer
   Format: TextHello, let me introduce myself to the nForum community with best wishes for a good and mathematically exciting start into the new year 2025 with a bon-mot in german " Mathematiker sind Maschinen, die Kaffee in Theoreme verwandeln, darum gilt's im neuen Jahr wie im alten: die Mass Kaffee halten" I want to initiate a discussion about a fundamental question , that arises for each working mathematician from Kurt Goedels Incompleteness theorem. Who does give guarantees, that a certain, specified conjecture that I want to solve or to disprove is decideable at all? Most of the working mathematicians in gemeotry think, "That does not really happen and is music taking from future". I want to take up this question here and it is my intention to show, that such nondecidable mathematical questions (conjectures) really occur in working mathematicians every day life. It is well known that the question, whether the locally convex topological vector space \mathhbb R^ J with the product topology is realcompact, where J is an infinite coordinate set of inaccesable cardinality, is not decidable- a real Goedel phenomenon, the so called Borsuk-Ulam-phenomenon. A topological manifold can locally be embedded into \mathbb R^n. If I define an infinite dimensional topological manifold as a topological space (X, T_X) such that there exists an open Covering X=\bigcup_{i\in I}U_i, such that the topology on U_i is the induced topology of R^ J via a topological embedding U_i\hookrightarrow \mathbb R^ J , that is the straight forward generalization of a classical finite dimensional manifold via charts , where the cardinality of J is an inaccessible cardinal, then it is a nondecidable question whether the topological space (X, T_X) is locally realcompact , i.e. whether each point P\in X posesses a neighbourhood, that is realcompact. Local realcompactness is the correct generalization of local compactness in the finite manifold case. That means that this fundamental and elementary question , whether an infinite topological locally euclidean space is locally realcompact or not, is nondecidable. Can someone give an explicit example of such an infinite dimensional manifold ? That means that this is a "point of no return in geometry " unless we introduce a new axiom to infinite dimensional manifold theory in the same way, as mathematicians in ancient times had to introduce new euclidean or noneuclidean axioms. The question whether finite dimensional topological manifold theory is complete or incomplete in Kurt Goedels sense of mathematical logic is a question , that is much more close to our geometric imagination and intuition and to our structure of human mind is the question that I want to introduce into the nforum -discussion. The discussion is opened. Best, Hermann Sommer

   Hello,
   let me introduce myself to the nForum community with best wishes for a good and mathematically exciting start into the new year 2025 with a bon-mot in german " Mathematiker sind Maschinen, die Kaffee in Theoreme verwandeln, darum gilt's im neuen Jahr wie im alten: die Mass Kaffee halten"
   I want to initiate a discussion about a fundamental question , that arises for each working mathematician from Kurt Goedels Incompleteness theorem. Who does give guarantees, that a certain, specified conjecture that I want to solve or to disprove is decideable at all? Most of the working mathematicians in gemeotry think, "That does not really happen and is music taking from future".
   I want to take up this question here and it is my intention to show, that such nondecidable mathematical questions (conjectures) really occur in working mathematicians every day life.
   It is well known that the question, whether the locally convex topological vector space \mathhbb R^ J with the product topology is realcompact, where J is an infinite coordinate set of inaccesable cardinality, is not decidable- a real Goedel phenomenon, the so called Borsuk-Ulam-phenomenon.
   A topological manifold can locally be embedded into \mathbb R^n. If I define an infinite dimensional topological manifold as a topological space (X, T_X) such that there exists an open Covering X=\bigcup_{i\in I}U_i, such that the topology on U_i is the induced topology of R^ J via a topological embedding U_i\hookrightarrow \mathbb R^ J , that is the straight forward generalization of a classical finite dimensional manifold via charts , where the cardinality of J is an inaccessible cardinal, then it is a nondecidable question whether the topological space (X, T_X) is locally realcompact , i.e. whether each point P\in X posesses a neighbourhood, that is realcompact.
   Local realcompactness is the correct generalization of local compactness in the finite manifold case. That means that this fundamental and elementary question , whether an infinite topological locally euclidean space is locally realcompact or not, is nondecidable. Can someone give an explicit example of such an infinite dimensional manifold ?
   That means that this is a "point of no return in geometry " unless we introduce a new axiom to infinite dimensional manifold theory in the same way, as mathematicians in ancient times had to introduce new euclidean or noneuclidean axioms.
   The question whether finite dimensional topological manifold theory is complete or incomplete in Kurt Goedels sense of mathematical logic is a question , that is much more close to our geometric imagination and intuition and to our structure of human mind is the question that I want to introduce into the nforum -discussion.
   The discussion is opened.
   Best,
   Hermann Sommer
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 12th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Hermann Sommer, just to say that to properly display mathematical formulas in discussion here, just enclose the expressions in dollar signs as usual in `LaTeX` (and be sure to choose below the edit pane the option "Markdown+Itex" instead of just "Text".) For convenience of other readers, I am reproducing your message below, with some formatting added: *** *** Hello, let me introduce myself to the nForum community with best wishes for a good and mathematically exciting start into the new year 2025 with a bon-mot in german > Mathematiker sind Maschinen, die Kaffee in Theoreme verwandeln, darum gilt's im neuen Jahr wie im alten: die Mass Kaffee halten I want to initiate a discussion about a fundamental question , that arises for each working mathematician from Kurt Goedels Incompleteness theorem. Who does give guarantees, that a certain, specified conjecture that I want to solve or to disprove is decideable at all? Most of the working mathematicians in gemeotry think, "That does not really happen and is music taking from future". I want to take up this question here and it is my intention to show, that such nondecidable mathematical questions (conjectures) really occur in working mathematicians every day life. It is well known that the question, whether the locally convex topological vector space $\mathhbb{R}^J$ with the product topology is realcompact, where $J$ is an infinite coordinate set of inaccesable cardinality, is not decidable- a real Goedel phenomenon, the so called Borsuk-Ulam-phenomenon. A topological manifold can locally be embedded into $\mathbb{R}^n$. If I define an infinite dimensional topological manifold as a topological space $(X, T_X)$ such that there exists an open Covering $X=\bigcup_{i\in I}U_i$, such that the topology on $U_i$ is the induced topology of $\mathbb{R}^J$ via a topological embedding $U_i \hookrightarrow \mathbb{R}^J$, that is the straight forward generalization of a classical finite dimensional manifold via charts , where the cardinality of $J$ is an inaccessible cardinal, then it is a nondecidable question whether the topological space $(X, T_X)$ is locally realcompact, i.e. whether each point $P\in X$ posesses a neighbourhood, that is realcompact. Local realcompactness is the correct generalization of local compactness in the finite manifold case. That means that this fundamental and elementary question, whether an infinite topological locally euclidean space is locally realcompact or not, is nondecidable. Can someone give an explicit example of such an infinite dimensional manifold ? That means that this is a "point of no return in geometry" unless we introduce a new axiom to infinite dimensional manifold theory in the same way, as mathematicians in ancient times had to introduce new euclidean or noneuclidean axioms. The question whether finite dimensional topological manifold theory is complete or incomplete in Kurt Goedels sense of mathematical logic is a question, that is much more close to our geometric imagination and intuition and to our structure of human mind is the question that I want to introduce into the nforum -discussion. The discussion is opened. Best, Hermann Sommer *** ***

   Hi Hermann Sommer,

   just to say that to properly display mathematical formulas in discussion here, just enclose the expressions in dollar signs as usual in LaTeX (and be sure to choose below the edit pane the option “Markdown+Itex” instead of just “Text”.)

   For convenience of other readers, I am reproducing your message below, with some formatting added:

   ---

   ---

   Hello,

   let me introduce myself to the nForum community with best wishes for a good and mathematically exciting start into the new year 2025 with a bon-mot in german

   Mathematiker sind Maschinen, die Kaffee in Theoreme verwandeln, darum gilt’s im neuen Jahr wie im alten: die Mass Kaffee halten

   I want to initiate a discussion about a fundamental question , that arises for each working mathematician from Kurt Goedels Incompleteness theorem. Who does give guarantees, that a certain, specified conjecture that I want to solve or to disprove is decideable at all? Most of the working mathematicians in gemeotry think, “That does not really happen and is music taking from future”.

   I want to take up this question here and it is my intention to show, that such nondecidable mathematical questions (conjectures) really occur in working mathematicians every day life.

   It is well known that the question, whether the locally convex topological vector space with the product topology is realcompact, where is an infinite coordinate set of inaccesable cardinality, is not decidable- a real Goedel phenomenon, the so called Borsuk-Ulam-phenomenon.

   A topological manifold can locally be embedded into . If I define an infinite dimensional topological manifold as a topological space such that there exists an open Covering , such that the topology on is the induced topology of via a topological embedding , that is the straight forward generalization of a classical finite dimensional manifold via charts , where the cardinality of is an inaccessible cardinal, then it is a nondecidable question whether the topological space is locally realcompact, i.e. whether each point posesses a neighbourhood, that is realcompact.

   Local realcompactness is the correct generalization of local compactness in the finite manifold case. That means that this fundamental and elementary question, whether an infinite topological locally euclidean space is locally realcompact or not, is nondecidable. Can someone give an explicit example of such an infinite dimensional manifold ?

   That means that this is a “point of no return in geometry” unless we introduce a new axiom to infinite dimensional manifold theory in the same way, as mathematicians in ancient times had to introduce new euclidean or noneuclidean axioms.

   The question whether finite dimensional topological manifold theory is complete or incomplete in Kurt Goedels sense of mathematical logic is a question, that is much more close to our geometric imagination and intuition and to our structure of human mind is the question that I want to introduce into the nforum -discussion.

   The discussion is opened.

   Best, Hermann Sommer

   ---

   ---

3. • CommentRowNumber3.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeJan 13th 2025
   • (edited Jan 13th 2025)
   • PermaLink
   Author: Madeleine Birchfield
   Format: MarkdownItexHello, The n-Forum isn't very active as a discussion forum for mathematics in general. It's primary use is as a discussion forum for the nLab wiki, and most of it is just comments announcing that an edit has been made to the wiki, and much of the rest of the comments is discussion by the editors about the material inside the wiki articles. In the past, other users have come on the nForum and tried to start a discussion on a mathematical topic unrelated to the wiki, but have frequently received no response. There are other more active discussion forums for mathematics, such as the [Category Theory Zulip](https://categorytheory.zulipchat.com) with about 3400 members, where one might ask the same question and very likely get a discussion started on the topic.

   Hello,

   The n-Forum isn’t very active as a discussion forum for mathematics in general. It’s primary use is as a discussion forum for the nLab wiki, and most of it is just comments announcing that an edit has been made to the wiki, and much of the rest of the comments is discussion by the editors about the material inside the wiki articles. In the past, other users have come on the nForum and tried to start a discussion on a mathematical topic unrelated to the wiki, but have frequently received no response.

   There are other more active discussion forums for mathematics, such as the Category Theory Zulip with about 3400 members, where one might ask the same question and very likely get a discussion started on the topic.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere could and should be more discussion here. And has been in the past. Discussing on the nForum and making notes on the nLab should feed back into each other. That said, myself I don't have anything to offer regarding Hermann's post, but I encourage anyone who does to engage.

   There could and should be more discussion here. And has been in the past.

   Discussing on the nForum and making notes on the nLab should feed back into each other.

   That said, myself I don’t have anything to offer regarding Hermann’s post, but I encourage anyone who does to engage.