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RECENT POSTS IN THIS TOPIC

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FQXi BLOGS

May 21, 2019

image: brewbetter |

Prof. Hofstadter:

1. Consider "The set of all sets which Douglas Hofstadter is considering at the moment." How many elements does this set contain?

The answer would seem to be One (assuming you weren't in the middle of a discussion of any other set); namely, itself. And yet, whenever you stop reading this post and turn to other things, the answer will be Zero. Unless you pause to reflect on the answer having switched to Zero, in which case, it will be One.

A quick modification universalizes the situation: let NOW be "the set of all sets which are being considered at the moment." This set contains *at least* One element--itself--but at some future point, may contain Zero elements. (Nor is the whole construction self-erasing, as NOW still exists when no one is considering it; it just contains nothing.)

Indeed, there's an oddly waveform-like quality to such sets, such that they cannot ever be "observed" to contain no elements; and yet, by inference, we can know that, in their "unobserved" state, they must.

But how can a mathematical object--which many would regard as unchanging, static, true entities; a vision, for Platonists (of whom Gödel was one), of eternal and necessary relationships--have this Copenhagen-esque quality?

Can the fact of inquiry--interaction; the forming of information--effect not only physical systems, as in waveform collapse, but mathematical truths as well?

(Indeed, if the cosmos is mathematically reductive, must this not be the case? For in that world, physical phenomena are the same thing as math, so a mirroring effect will apply: what is true of photons must be true, at some suitably basic level, of mathematics.)

2. While we are chatting, consider "The set of all sets which are being considered today and only today." (Plenty of sets are under consideration today, but none of them are members of this set, call it TDY, because they have all been considered on previous days as well. Even if a logician somewhere has just constructed a new set, that set does not belong in TDY, as it will be under consideration from now on.)

TDY would likewise seem to contain One element today--itself--and Zero elements tomorrow. But matters are stranger for TDY than for NOW. Because when you wake up tomorrow, Prof. Hofstadter, and "check," there will be no such set as TDY (that set you will be thinking about when you check, you also thought about yesterday, which means it isn't that set). It's not that TDY will be empty, or even that it will have ceased to exist as a set; it cannot even be thought about.

The very "eternally true" nature of mathematical objects, it seems, leads paradoxically to this one's disappearance, as it will block us even from saying that TDY *was* the set under consideration yesterday. That set cannot be TDY, by definition; and, since mathematical objects are nothing other than their logical definitions, the eternal set TDY both has and does not have an existence.

Mere word games? Perhaps. (Or concept games.) Most readers will quickly dismiss such ideas out of hand: "You can't define a set that way." But is such a contrivance any less valid than Bertrand Russell's "The Set of All Sets Which Don't Include Themselves As Members"? The later convolution was constructed exactly to pursue the inherent limits of logical definition, and its famous resulting paradox ("Does this set contain itself?") revealed the incompletion of Cantor's set theory.

FQXi fans await your reply.

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Mr. Orem,

Hi. In regard to the below section in your blog post:

"Consider "The set of all sets which Douglas Hofstadter is considering at the moment." How many elements does this set contain? The answer would seem to be One (assuming you weren't in the middle of a discussion of any other set); namely, itself."

my view is that a set doesn't even exist until the list of elements it contains is completely defined. So, consider a set, R, which is the set of all sets being considered by someone at time t=1. All those sets being considered at t=1 are the elements of R. But, this list of elements in R isn't complete until after t=1. This means that if a set doesn't exist until the list of elements it contains is completely defined, then set R doesn't exist until after t=1. So, if R isn't completely defined and doesn't exist during t=1, it can't be a member of itself.

I think this has some implications for Russell's Paradox. Thank you.

Roger Granet

https://sites.google.com/site/ralphthewebsite/filecabi

net/theory-russell-paradox-godel

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Hi. In regard to the below section in your blog post:

"Consider "The set of all sets which Douglas Hofstadter is considering at the moment." How many elements does this set contain? The answer would seem to be One (assuming you weren't in the middle of a discussion of any other set); namely, itself."

my view is that a set doesn't even exist until the list of elements it contains is completely defined. So, consider a set, R, which is the set of all sets being considered by someone at time t=1. All those sets being considered at t=1 are the elements of R. But, this list of elements in R isn't complete until after t=1. This means that if a set doesn't exist until the list of elements it contains is completely defined, then set R doesn't exist until after t=1. So, if R isn't completely defined and doesn't exist during t=1, it can't be a member of itself.

I think this has some implications for Russell's Paradox. Thank you.

Roger Granet

https://sites.google.com/site/ralphthewebsite/filecabi

net/theory-russell-paradox-godel

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William, you have written "let NOW be "the set of all sets which are being considered at the moment."" That NOW sounds like the experienced present to me. Further on you have written "(Nor is the whole construction self-erasing,as NOW still exists when no one is considering it; it just contains nothing.)"

Does NOW, the experienced present, still exist when it is not being thought about/...

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Does NOW, the experienced present, still exist when it is not being thought about/...

view entire post

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William, I had another thought. Is NOW just a three letter name assigned to the set of all sets being thought about at the moment, with no further implication? In that case the mathematical representation NOW can still exist when empty but what is the physical state that corresponds to the empty set? All minds empty of thoughts about sets.

Which brings me back to, 'is it a realistic scenario for no one, out of the many billions in the world, to be thinking about sets?'. Probably not which makes it always at least one set, not zero.

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Which brings me back to, 'is it a realistic scenario for no one, out of the many billions in the world, to be thinking about sets?'. Probably not which makes it always at least one set, not zero.

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Yes! The measure, or definition, of the set is like a quantum collapse of the wave function.

The wave function defines a probability, then we are sure of the set definition when we verify the proposition; some definition are circular, so there is a continuous change in the proposition value (true,false) and the proposition cannot be resolved (measure with too error? Or impossibility to measure under Plank lenght?).

I am thinking that the right approximation of the Russel problem is a quantum flip-flop: a system that for each quantum input give the orthogonal system (bi-stable: for example spin z ->spin x and spin x-> spin z), the output of the system is the measure, then if the input is the output of the system there is indetermiation. This is a measure system that give not a measure value, there is not a collapse of the wave function (but there is the collapse in each step). The truth value is 1/2.

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The wave function defines a probability, then we are sure of the set definition when we verify the proposition; some definition are circular, so there is a continuous change in the proposition value (true,false) and the proposition cannot be resolved (measure with too error? Or impossibility to measure under Plank lenght?).

I am thinking that the right approximation of the Russel problem is a quantum flip-flop: a system that for each quantum input give the orthogonal system (bi-stable: for example spin z ->spin x and spin x-> spin z), the output of the system is the measure, then if the input is the output of the system there is indetermiation. This is a measure system that give not a measure value, there is not a collapse of the wave function (but there is the collapse in each step). The truth value is 1/2.

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I am thinking that if the axioms of the reality are the sensors (for example the sensor - ensemblr of experiment evidence - for: the Earth is round), and there is a net of analogic circuit (it is not necessary the quantum circuit), then the existence of the flip-flop is equivalent to the existence of indecidibil propositions (the circuit don't give a stable state, with yes or not).

It is equivalent to the Godel's incompleteness theorems, or Church, or Turing theories.

If it is accepted an intermediate response, with a quantum calculus (if there is overlapping, without collapse, between the input and output function in the quantum analog circuit), then there is the possibility to solve the problem of the logic, associating a response with a probability; but so nothing is more sure, only probable.

this post has been edited by the author since its original submission

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It is equivalent to the Godel's incompleteness theorems, or Church, or Turing theories.

If it is accepted an intermediate response, with a quantum calculus (if there is overlapping, without collapse, between the input and output function in the quantum analog circuit), then there is the possibility to solve the problem of the logic, associating a response with a probability; but so nothing is more sure, only probable.

this post has been edited by the author since its original submission

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"(Indeed, if the cosmos is mathematically reductive, must this not be the case? For in that world, physical phenomena are the same thing as math..."

"if the cosmos... where physical phenomena are the same thing as math", actually existed, one would be able to survive being lost in a hot desert, by drinking only math, since water would not be necessary. Obviously, the cosmos is not such a place. Math and Physics are not one and the same thing - as discussed in my FQXI essay, last year.

Rob McEachern

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"if the cosmos... where physical phenomena are the same thing as math", actually existed, one would be able to survive being lost in a hot desert, by drinking only math, since water would not be necessary. Obviously, the cosmos is not such a place. Math and Physics are not one and the same thing - as discussed in my FQXI essay, last year.

Rob McEachern

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I think that the reality that exists externally and independently of the mind can be described using mathematical relationships. The mind (product of brain activity) is not merely a part of that aforementioned reality though. The mind fabricates a reality that is qualitatively different. One difference being objects, or sets, can disappear, i.e. not exist, and reappear, i.e. come back into existence, in a way that actual objects and sets external to, and independent of, the mind can not. So actual objects, and sets, can not be treated as equivalent mathematical objects to objects and sets that are products of the mind.

Let me try to say that more clearly. An object or set in mind/s is not equivalent to an object or set of the same name and description that exists external and independently of mind/s. There are qualitative differences.

The apple imagined and the image of an apple fabricated by the mind from received data are not the same as an actual apple sitting in the fruit bowl over there, separate from the human mind. Though both are called 'apple' and might be verbally described in the same way. One is made of electrical and chemical activity in a brain and the other is made of molecules.

Seems to me there is a similarity with wave function collapse as there is a switch from probabilities being thought about to a singular measured outcome being thought about. So the many possibilities cease to exist for the observer, like the set not thought about, discussed in the article.

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Let me try to say that more clearly. An object or set in mind/s is not equivalent to an object or set of the same name and description that exists external and independently of mind/s. There are qualitative differences.

The apple imagined and the image of an apple fabricated by the mind from received data are not the same as an actual apple sitting in the fruit bowl over there, separate from the human mind. Though both are called 'apple' and might be verbally described in the same way. One is made of electrical and chemical activity in a brain and the other is made of molecules.

Seems to me there is a similarity with wave function collapse as there is a switch from probabilities being thought about to a singular measured outcome being thought about. So the many possibilities cease to exist for the observer, like the set not thought about, discussed in the article.

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"I think that the reality that exists externally and independently of the mind can be described using mathematical relationships."

I agree.

"The mind fabricates a reality that is qualitatively different."

Again, I agree. But I would point-out the key-word : "fabricated". It is indeed fabricated, and it is quantitatively different in addition to being qualitatively...

view entire post

I agree.

"The mind fabricates a reality that is qualitatively different."

Again, I agree. But I would point-out the key-word : "fabricated". It is indeed fabricated, and it is quantitatively different in addition to being qualitatively...

view entire post

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Thank you Robert,

-Rather than... can not..., I should have said -So actual objects, and sets, *should* not be treated as equivalent mathematical objects to objects and sets that are products of the mind.

You are correct it -is- done and incidentally that's why we have the barn pole paradox.

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-Rather than... can not..., I should have said -So actual objects, and sets, *should* not be treated as equivalent mathematical objects to objects and sets that are products of the mind.

You are correct it -is- done and incidentally that's why we have the barn pole paradox.

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Dear All,

I is the set of sets and the mathematical truth is zero = I = infinity.

I determines the number of sets I wants to see at any given moment.

Universe is an iSphere.

Love,

Sridattadev.

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I is the set of sets and the mathematical truth is zero = I = infinity.

I determines the number of sets I wants to see at any given moment.

Universe is an iSphere.

Love,

Sridattadev.

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Whether it is an imaginary apple, an image of an apple, or an actual apple, it is structure. It forms, exists, dissolves/is eaten. The energy which manifested it, whether fructose, neurons or pixels, continues on as other forms and is conserved in a dynamic presence, while the temporal existence of that particular shape fades into the past. In a four dimensional math, where the passage of time is an illusion, where is the math to explain why you can't have your apple and eat it too?

It time an illusion, or is does the math create a bias toward stasis that is not very reflective of reality?

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It time an illusion, or is does the math create a bias toward stasis that is not very reflective of reality?

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Is time an illusion, or does the math create a bias toward stasis that is not reflective of reality?

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John,

"Why can't you have your apple and eat it too?" I like it. May be we can. Isn't Newtonian mechanics with 3 dimensions and passage of time dealing with actual objects in space and Einstein's relativity dealing with appearances, space-time images? The time part of space-time is an integrated part of the illusion but passage of time is tied to sequential changes within space, the fundamental reality.

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"Why can't you have your apple and eat it too?" I like it. May be we can. Isn't Newtonian mechanics with 3 dimensions and passage of time dealing with actual objects in space and Einstein's relativity dealing with appearances, space-time images? The time part of space-time is an integrated part of the illusion but passage of time is tied to sequential changes within space, the fundamental reality.

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John,

You asked: "Is time an illusion, or does the math create a bias toward stasis that is not reflective of reality? "

Time is not an illusion. And the math does not create a bias that is not reflective of reality.

But the faulty "interpretations" of the math, fabricated entirely within the minds of physicists, *has* created such a bias. Specifically, the equations of physics seem to work the same, going either forward, or backward in time. It is only the vast information content of the initial conditions, that breaks this symmetry. Consequently, when you only examine the low-information-content equations, and ignore the high-information-content initial conditions, you are prone to fabricate a view of reality that bears little resemble to that reality. The math still fits the data, but the bad interpretations of the "meaning" of the math do not fit anything; hence all the purported weirdness in the "interpretations" of modern physics.

Rob McEachern

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You asked: "Is time an illusion, or does the math create a bias toward stasis that is not reflective of reality? "

Time is not an illusion. And the math does not create a bias that is not reflective of reality.

But the faulty "interpretations" of the math, fabricated entirely within the minds of physicists, *has* created such a bias. Specifically, the equations of physics seem to work the same, going either forward, or backward in time. It is only the vast information content of the initial conditions, that breaks this symmetry. Consequently, when you only examine the low-information-content equations, and ignore the high-information-content initial conditions, you are prone to fabricate a view of reality that bears little resemble to that reality. The math still fits the data, but the bad interpretations of the "meaning" of the math do not fit anything; hence all the purported weirdness in the "interpretations" of modern physics.

Rob McEachern

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Hi William,

Once again, you have brought into the light one of the most profound questions at the intersection of mathematics and physics:

"Can the fact of inquiry--interaction; the forming of information--effect not only physical systems, as in waveform collapse, but mathematical truths as well?"

Did you really mean 'effect' as in "to bring into being," or 'affect,' as in "to alter a prior condition?"

The conventional interpretation of quantum mechanics largely prefers the former meaning; a discrete observation -- a measurement -- creates a reality that was not there before. If the meaning "affect" is applied, one presumes a continuous reality that is continuous not only in terms of spacetime; it is continuous with the discrete mathematics that describes both the nature of spacetime and the changing relative states of the mass-energy points that are part of the continuum.

The latter is closer to both Einstein's relativity and Wheeler's participatory universe. That is, discrete sets are the results of an evolving continuous reality, not the cause of it.

Tom

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Once again, you have brought into the light one of the most profound questions at the intersection of mathematics and physics:

"Can the fact of inquiry--interaction; the forming of information--effect not only physical systems, as in waveform collapse, but mathematical truths as well?"

Did you really mean 'effect' as in "to bring into being," or 'affect,' as in "to alter a prior condition?"

The conventional interpretation of quantum mechanics largely prefers the former meaning; a discrete observation -- a measurement -- creates a reality that was not there before. If the meaning "affect" is applied, one presumes a continuous reality that is continuous not only in terms of spacetime; it is continuous with the discrete mathematics that describes both the nature of spacetime and the changing relative states of the mass-energy points that are part of the continuum.

The latter is closer to both Einstein's relativity and Wheeler's participatory universe. That is, discrete sets are the results of an evolving continuous reality, not the cause of it.

Tom

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Tom,

I don't think the issue is quite so deep and/or complicated.

In the statement "The set of all sets which Douglas Hofstadter is considering at the moment.",...

How does the use of the word "considering" differ from mentally "observing", as the latter word is used in the statement: "Indeed, there's an oddly waveform-like quality to such sets, such that they cannot ever be "observed" to contain no elements; and yet, by inference, we can know that, in their "unobserved" state, they must." ?

The set being considered is the observed set. The number of elements in this set is not defined to be a constant, nor is it required to be equal to the number of elements in an unobserved/unconsidered set.

It is ultimately no different than asking why the number of elements (molecules) of water, in an empty glass, differs from the number that may exist in a glass that was not defined to be empty. One glass/set is in the state "empty" *by definition*, the state of the other is *undefined*, and can only be determined by an act of "considering" or "observing". Attempting to compare such dissimilar types of entities, is a "category" mistake. The state of one is "contingent" upon some action. The state of the other is not contingent upon anything; unlike the former, it is what it is, by definition.

Rob McEachern

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I don't think the issue is quite so deep and/or complicated.

In the statement "The set of all sets which Douglas Hofstadter is considering at the moment.",...

How does the use of the word "considering" differ from mentally "observing", as the latter word is used in the statement: "Indeed, there's an oddly waveform-like quality to such sets, such that they cannot ever be "observed" to contain no elements; and yet, by inference, we can know that, in their "unobserved" state, they must." ?

The set being considered is the observed set. The number of elements in this set is not defined to be a constant, nor is it required to be equal to the number of elements in an unobserved/unconsidered set.

It is ultimately no different than asking why the number of elements (molecules) of water, in an empty glass, differs from the number that may exist in a glass that was not defined to be empty. One glass/set is in the state "empty" *by definition*, the state of the other is *undefined*, and can only be determined by an act of "considering" or "observing". Attempting to compare such dissimilar types of entities, is a "category" mistake. The state of one is "contingent" upon some action. The state of the other is not contingent upon anything; unlike the former, it is what it is, by definition.

Rob McEachern

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Rob,

You're ignoring the fundamental question. Does the act of observation (measure) actually create a discrete physical phenomenon, or describe discrete change in a continuous physical phenomenon?

You accept that observing is identical to creating, by invoking a linguistic kludge (" ... the state of the other is *undefined*, and can only be determined by an act of 'considering' or 'observing'.) I think the actual physics is more elegant, and favors the continuum. Considering observing is a potential; it is not an act of observing, which is embedded in the continuum.

Tom

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You're ignoring the fundamental question. Does the act of observation (measure) actually create a discrete physical phenomenon, or describe discrete change in a continuous physical phenomenon?

You accept that observing is identical to creating, by invoking a linguistic kludge (" ... the state of the other is *undefined*, and can only be determined by an act of 'considering' or 'observing'.) I think the actual physics is more elegant, and favors the continuum. Considering observing is a potential; it is not an act of observing, which is embedded in the continuum.

Tom

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Tom, Rob,

We do need to extract our perception of reality from the continuum. Consider the function of photography. The continuum of light must be filtered, focused, timed, the aperture set, lighting considered, position selected, etc. Otherwise you just have white light. It is in many ways the same process used to measure quantum behavior and how the results reflect our measuring devices.

Leave the shutter open a little longer and you have "momentum." Shorter and you have "position." You can't do both in the same picture.

It is balancing many dichotomies; Content and context. Dynamic and static. Distinction and connection. Energy and information. Continuum and discretion. Focused and distributed. Etc.

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We do need to extract our perception of reality from the continuum. Consider the function of photography. The continuum of light must be filtered, focused, timed, the aperture set, lighting considered, position selected, etc. Otherwise you just have white light. It is in many ways the same process used to measure quantum behavior and how the results reflect our measuring devices.

Leave the shutter open a little longer and you have "momentum." Shorter and you have "position." You can't do both in the same picture.

It is balancing many dichotomies; Content and context. Dynamic and static. Distinction and connection. Energy and information. Continuum and discretion. Focused and distributed. Etc.

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My daughter told me last night -- my 7-year-old granddaughter said to her, "Mom, if numbers go on forever, all numbers are small." Absolutely true.

Shades of Leibniz, who believed that the foundations of nature could be understood in the behavior of the infinitely small (Hermann Weyl commented on it in *The Philosophy of Mathematics and Natural Science*). So I thought of Thomson's lamp...

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Shades of Leibniz, who believed that the foundations of nature could be understood in the behavior of the infinitely small (Hermann Weyl commented on it in *The Philosophy of Mathematics and Natural Science*). So I thought of Thomson's lamp...

view entire post

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Tom,

"... all numbers are small." Absolutely true."? Was Galileo really less intelligent than your granddaughter? I rather accept his argument that smaller than, equal to, and larger than must not be applied on infinite quantities.

Are there actually more rational than real numbers? Isn't being infinite just a property even if we often may benefit from using it as if it was a number?

May we not unmask G. Cantor's idea of transfinite numbers from aleph_2 on as overly naive? Didn't Ebbinghaus quote Lessing as to speak of an error?

Is it at all allowed without any precaution to "relate finite points to infinite sets of points"? What is an infinite point if a point is according to Euclid something that has no parts?

I don't see a problem to understand the limit of the infinite sum you mentioned (with 0, not -1, for unlit). When Cantor was asked to justify his position in front of the same apparent dilemma, he failed to give a convincing answer.

In all, when I yesterday mentioned the mutilation of mathematics, I intended referring to the de facto ban of the genuine continuum and what Cantor called the infinitum absolutum from a realm that only admits pebble-like numbers.

A second flaw of 19th century was perhaps even worse: The old idea that the reality is built on mathematics was not consequently abandoned when mathematics was declared an independent discipline. I call this abuse of mathematics.

Eckard

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"... all numbers are small." Absolutely true."? Was Galileo really less intelligent than your granddaughter? I rather accept his argument that smaller than, equal to, and larger than must not be applied on infinite quantities.

Are there actually more rational than real numbers? Isn't being infinite just a property even if we often may benefit from using it as if it was a number?

May we not unmask G. Cantor's idea of transfinite numbers from aleph_2 on as overly naive? Didn't Ebbinghaus quote Lessing as to speak of an error?

Is it at all allowed without any precaution to "relate finite points to infinite sets of points"? What is an infinite point if a point is according to Euclid something that has no parts?

I don't see a problem to understand the limit of the infinite sum you mentioned (with 0, not -1, for unlit). When Cantor was asked to justify his position in front of the same apparent dilemma, he failed to give a convincing answer.

In all, when I yesterday mentioned the mutilation of mathematics, I intended referring to the de facto ban of the genuine continuum and what Cantor called the infinitum absolutum from a realm that only admits pebble-like numbers.

A second flaw of 19th century was perhaps even worse: The old idea that the reality is built on mathematics was not consequently abandoned when mathematics was declared an independent discipline. I call this abuse of mathematics.

Eckard

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Eckard,

"'... all numbers are small.' Absolutely true."?

Absolutely true. Fortunately, I tracked down the actual words she said, as my daughter relayed in a text message: "Mom, if numbers just keep on going and going on forever, that means that all numbers are small."

It's an elegant and useful way to say that infinity is not a number.

"Was Galileo really less intelligent than your granddaughter?"

Of course. Just as I expect some significant proportion of 7-year-olds several hundred years from now to be more intelligent than Einstein. Evolution is necessarily progressive if new knowledge contributes to survival and growth of the species. The distribution of species intelligence is relative to the age, as individual intelligence is relative to the Gaussian distribution of the group measured.

"I rather accept his argument that smaller than, equal to, and larger than must not be applied on infinite quantities."

Quantities are not infinite if infinity is not a number, are they?

Tom

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"'... all numbers are small.' Absolutely true."?

Absolutely true. Fortunately, I tracked down the actual words she said, as my daughter relayed in a text message: "Mom, if numbers just keep on going and going on forever, that means that all numbers are small."

It's an elegant and useful way to say that infinity is not a number.

"Was Galileo really less intelligent than your granddaughter?"

Of course. Just as I expect some significant proportion of 7-year-olds several hundred years from now to be more intelligent than Einstein. Evolution is necessarily progressive if new knowledge contributes to survival and growth of the species. The distribution of species intelligence is relative to the age, as individual intelligence is relative to the Gaussian distribution of the group measured.

"I rather accept his argument that smaller than, equal to, and larger than must not be applied on infinite quantities."

Quantities are not infinite if infinity is not a number, are they?

Tom

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No Tom,

While your granddaughter certainly understood that any natural number is larger than its predecessor, she did perhaps not understand in what sense she used the notion "all" numbers and that small always needs a reference. She might have heard from someone that any number is smaller than infinity or, in other words, small in comparison with infinity, which is questionable because infinity is no measure that can be increased or decreased: Infinity plus or minus any number is still infinity. I am not using Leibniz's notion of relative infinity.

Galileo used what we are calling bijection as to show that there are not more natural numbers 1, 2, 3, 4, 5, ... than their squares 1, 4, 9, 16, 25, ...

When Galileo's Salviati spoke of infinite quantities, he meant such infinite series. I consider Galileo's reasoning flawless and convincing while G. Cantor's infinitum creatum sive Transfinitum was obviously based on naivety.

Your granddaughter might be more intelligent than G. Cantor and Einstein. However, I doubt that their mental ability will come close to Galileo's.

Eckard

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While your granddaughter certainly understood that any natural number is larger than its predecessor, she did perhaps not understand in what sense she used the notion "all" numbers and that small always needs a reference. She might have heard from someone that any number is smaller than infinity or, in other words, small in comparison with infinity, which is questionable because infinity is no measure that can be increased or decreased: Infinity plus or minus any number is still infinity. I am not using Leibniz's notion of relative infinity.

Galileo used what we are calling bijection as to show that there are not more natural numbers 1, 2, 3, 4, 5, ... than their squares 1, 4, 9, 16, 25, ...

When Galileo's Salviati spoke of infinite quantities, he meant such infinite series. I consider Galileo's reasoning flawless and convincing while G. Cantor's infinitum creatum sive Transfinitum was obviously based on naivety.

Your granddaughter might be more intelligent than G. Cantor and Einstein. However, I doubt that their mental ability will come close to Galileo's.

Eckard

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William Orems’ open letter to Douglas Hofstadter contains a delicious demonstration of self-referential pitfalls, constructed with the help of some hitherto unknown (at least to me) set definitions. His variations over the theme of the “sets of all sets” are in good tradition with the results and examples given in the past by such famous logicians like Bertrand Russell or Kurt Gödel, who...

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Hofstadter wrote:

"... there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive "upward" shifts turn out to give rise to a closed cycle. That is, despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop."

I also agree with Orem when he concluded:

"... revealed the incompletion of Cantor's set theory."

Isn't the belief in physically real singularities at variance with the notion of a continuum every part of which has parts?

If Goedel was not "the most doggedly fastidious individual" but in position to question Cantor's finitism, then there was perhaps no Neumann-Goedel-Bernays alternative to Principia Mathematica and ZFC.

Goedel planned a second part of his 1930 paper as to establish a link between consistency proofs and type theory, but he did not publish a second part of that paper before his death: "Ueber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes (1958).

Can we hope for curing such foundational flaws just by acupuncture and humor as suggested by Zenkin?

Eckard Elumschein

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"... there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive "upward" shifts turn out to give rise to a closed cycle. That is, despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop."

I also agree with Orem when he concluded:

"... revealed the incompletion of Cantor's set theory."

Isn't the belief in physically real singularities at variance with the notion of a continuum every part of which has parts?

If Goedel was not "the most doggedly fastidious individual" but in position to question Cantor's finitism, then there was perhaps no Neumann-Goedel-Bernays alternative to Principia Mathematica and ZFC.

Goedel planned a second part of his 1930 paper as to establish a link between consistency proofs and type theory, but he did not publish a second part of that paper before his death: "Ueber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes (1958).

Can we hope for curing such foundational flaws just by acupuncture and humor as suggested by Zenkin?

Eckard Elumschein

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