"The universe can and will create itself from nothing."

Faith & Science Conversation
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ChatGPT, can you make an argument for why infinity is a non-numerical value?

Yes, I can make an argument for why infinity is a non-numerical value.

Infinity is often used to represent a quantity that is unbounded or undefined. It is not a number in the traditional sense because it does not represent a specific quantity. Instead, it represents a concept of endlessness or boundlessness that cannot be quantified.

For example, consider the limit of a function as x approaches infinity. This limit does not represent a specific numerical value but rather the behavior of the function as x gets larger and larger. Similarly, in geometry, infinity can represent a point or a line that extends indefinitely, but it does not have a specific numerical value.

Moreover, infinity is not included in the traditional set of numbers, such as the natural numbers, integers, or real numbers, because it violates many of the properties of these sets. For instance, infinity cannot be added, subtracted, multiplied, or divided like regular numbers, and it does not have a well-defined order.

Therefore, due to its conceptual nature and inability to be quantified or operated on like regular numbers, infinity can be considered a non-numerical value.

Can you make an argument for why infinity is a numerical value?

Yes, I can make an argument for why infinity is a numerical value.

Infinity is a concept that is used in mathematics to represent a quantity that is unbounded or limitless. It is a number that is larger than any finite number and is often used in mathematical equations and calculations.

In some branches of mathematics, such as calculus and analysis, infinity is treated as a number and is used to represent the limit of a sequence or a function. The concept of infinity is also used in geometry to describe points, lines, and planes that extend indefinitely.

Furthermore, there are different types of infinity that can be described mathematically, such as countable and uncountable infinities. These infinities are used in set theory and are treated as numerical values.

In addition, infinity is often included in the set of real numbers, which are the numbers that can be represented on a number line. The real numbers include both finite numbers and infinite values, such as positive and negative infinity.

Therefore, due to its use in mathematical equations and calculations, its treatment as a number in certain branches of mathematics, and its inclusion in the set of real numbers, infinity can be considered a numerical value.

:grin: … Can you make an argument for why married bachelors exist?

No, I cannot make an argument for why married bachelors exist. The concept of a “married bachelor” is a logical contradiction, as the definition of a bachelor is an unmarried man, and the definition of marriage involves a formal union between two people. Therefore, it is impossible for a person to be both married and a bachelor at the same time.

This logical contradiction is similar to the concept of a “square circle.” Just as it is impossible for a shape to simultaneously have the properties of a square and a circle, it is impossible for a person to simultaneously have the properties of a married person and a bachelor.

In philosophy, the law of non-contradiction states that a proposition cannot be both true and false at the same time and in the same sense. The concept of a married bachelor violates this fundamental law of logic and is therefore impossible.

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How ironic that atheist worshippers of methodological naturalism can also believe in superstition and magic.

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Hawking the atheist is on record as saying,

“There is no heaven or afterlife … that is a fairy story for people afraid of the dark.”

Perhaps one could argue that atheists like Hawking are afraid of the light.

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It can be frightening even for those who love the light. I heard Craig Keener teaching on this passage from Revelation a year or so back and it stuck with me:

Everyone hid in caves and among the rocks of the mountains. This included the kings of the earth, the princes and the generals. It included rich people and powerful people. It also included everyone else, both slaves and people who were free. They called out to the mountains and rocks, “Fall on us! Hide us from the face of the one who sits on the throne! Hide us from the anger of the Lamb! The great day of their anger has come. Who can live through it?”

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Sorry I missed the Aquinas thread. I have limited bandwidth these days.

It’s beyond my paygrade, but the inventor of “set theory” Georg Cantor proposed multiple types of infinities, some larger than others:

Regarding an infinite number of points in an infinite space, Cantor also showed there are an infinite number of “real” numbers between 0 and 1. From the same article:
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So if I drew a 1 meter line between two points on the ground (or in space) and asked Mike to measure every real number along that line, he would never finish. As Cantor also showed, there are an infinite number of points (real numbers) within that 1 meter of space. You don’t need an infinite space to contain an infinite number of points. Zero to One will do.

I don’t think the conclusion follows from the premises. The Father, Son, and Holy Spirit existed prior to the hot Big Bang and prior to the rapid expansion that preceded it. Before that, we have nothing but speculation. Who knows how long the triune Godhead existed in fellowship and communication before initiating whatever “first cause” brought forth the universe as we know it?

As Cantor showed, there can be an infinite number of measurable points between the space of 0 and 1 meter in length. Surely humanity has instruments that can measure 0.00012 mm? What about 0.000012 mm? Just because our instruments eventually reach their limits doesn’t mean those numbers aren’t actual and can’t be delimited. I’m sure @pevaquark can explain the math far better than I can.

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106

ChatGPT isn’t a fact-finding algorithm. That’s a basic misunderstanding of the software, like GenZ using TikTok instead of Google.

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107

Hawking changed his mind in his “final theory.” Google is your friend. You’ll have to unpack your meaning in the rest. Care to elaborate?

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Interesting. Somehow it would be like saying a book is neither a source of facts. It did surprise me that it got the law of non-contradiction right in that a proposition cannot be A and non-A at the same time and in the same relationship.

Fact-determining is probably the better term for what you mean.

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I’ve written to you before about the two kinds of infinite values that I know of with the real and natural numbers. Which is the difference between (X<—>non-X) and (X,X+1,X+2…).

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Sorry that I was not clear about the points in space, I meant actual points, like stars in the sky. Not like virtual points that exist as part of a mathematical function.

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Please explain.

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I have a few objections to the analysis in this video “9 levels of nothing”.

  1. Fields, radiation and particles cannot be distinguished. These are just different descriptions of the same physical reality.

  2. Somewhere in this removal list you have to remove the box and the universe around it. Seeing no way to distinguish the vacuum from space-time and the laws of nature, they would be “removed” when those are “removed.”

  3. Then since the box and the universe is gone, talking about non-physical entities being in the box is a bit bizarre. Setting that aside, I don’t know why you would think they are in the so called box. I guess you would have to be specifically talking about entities which are omnipresent and cannot be excluded from anything unless they do not exist at all. The Christian God has been described in this way by some.

  4. Talking about removing abstractions like mathematics is even more bizarre… likely an artifact of Plato’s rhetoric which I think is total nonsense, treating such things as having an existence apart from the minds who formulate and think about them. At most you can say they are already removed along with minds such as God’s mind which may think about them.

  5. Possibilities are already removed as part of other removals: the possibilities which are a part of the laws of nature gone when those are removed (fields, for example, can be described as the possibility of a particle), and abstract possibilities removed when abstractions are removed with the minds that can think about them. I certainly deny the existence of any thing we can name possibilities apart from these other things.

So… at most that leaves me only 5 “levels of nothing.”

I. no visible things in the box
II. no air (or other nonvisible materials) in the box
III. no particles. fields, or radiation in the box
IV. no vacuum, space-time, or laws of nature → no box.
V. no (omnipresent or thinking) being to even think of the nonexistent box or any other abstractions.

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Of course, you are right in your objections. She really did move from the physical to the existential. (Her new book is called Existential Physics, which may have something to do with her thinking? I haven’t looked at the blurb yet.)

It was, however an engaging exercise in considering what we actually mean by “nothing” as well as how many of our own assumptions we bring along with us, when we consider any idea — even “nothing.”

So yes, in the end, the video was really a philosophical exercise in critical thinking rather than physics.

Good Lord, we need more of that!

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ChatGPT, are you a fact finding algorithm?

ChatGPT is not strictly a fact-finding algorithm, but it is a language model that has been trained on a large corpus of text data to generate human-like responses to a wide variety of prompts. While ChatGPT can provide factual information in its responses, it does not necessarily prioritize finding or verifying facts as its primary function. Instead, ChatGPT is designed to generate coherent and relevant responses that are similar to those that a human might provide in a given situation, based on the patterns and structures it has learned from the text it has been trained on.

What is an example of a fact finding algorithm?

A fact-finding algorithm is a program or software designed to gather and verify information or data from a variety of sources. It is typically used in fields such as journalism, research, and data analysis, where accuracy and reliability of information are crucial.

@Jay313 Based on this response from ChatGPT, I would like to say you are correct it is not a fact finding algorithm. But it is still a helpful and easily accessible tool for finding facts, clarifying ideas, and researching concepts.

Edit: this is curious, I reran the question and got this response,

ChatGPT, are you a fact finding algorithm?

Yes, as an AI language model, one of my primary functions is to find and present factual information on a wide range of topics. However, my abilities go beyond just finding facts - I can also engage in conversations, generate text based on prompts, and perform various language-related tasks such as language translation and summarization.

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The above quote is correct about it being ‘boundless’ and unquantifiable - and we could even probably make the case for ‘undefined’ since that is the accepted language of most math texts regarding infinity - though one could quibble (as I am about to) on that last point. I think it is increasingly well defined as a concept - which is where it is useful. It is trivial to see that of course it can’t be represented by any definite real number - hence not being one. But we can distinguish between different kinds of infinities as others have also already noted above. Rather than simply noting that an answer is “Undefined” (which some textbooks are okay with), I tell my calculus students that they can give more specific and informative answers. E.g. Is this function approaching positive infinity or negative infinity? Or in what manner is it proceeding off to infinity - is it going in a linear sense (an angled asymptote?) or is it ‘asymptotically’ approaching a parabolic or some other kind of curve? Those are questions that help students visualize ‘end behavior’ of functions.

I think the above assertion is just plain wrong. Infinite limits can and do approach very precisely defined real numbers.

E.g. The limit as x approaches infinity of the quotient: 3x / (5x - 2sin x) is exactly and precisely 3/5.
Limits are infinitely precise that way - and quite well defined as such.

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114

Agreed

That’s a good way to say it. An actually infinite quantity of things, would be like having a genuinely unquantifiable quantity. Not because the observer is temporally or spatially limited, but because the collection naturally exists as such.

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I was just trying to grow my understanding of the aleph numbers (beyond aleph null) and found this video a fascinating watch. I would still like to see if somebody can just provide examples of aleph 1, or aleph 2, etc - e.g. If the ‘uncountable’ set of reals between zero and one is an example of an aleph 1 set … is there a good similar way to visualize an aleph 2 set (perhaps, say, the set of points on a 1x1 square of the plane adjacent to the number line?) Because at least the reals on the number line between zero and one are … traversible? … (is that a thing? It should be…because one could ‘move through’ every one of those uncountables without skipping over any). But one can do no such thing with a plane. Anyway - none of this really gets at what they were discussing of the aleph numbers - so I still have a ways to go … so to speak.

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I’d love to see a couple people who are knowledgeable in this subject have a discussion. I would be taking notes and asking questions.

From the little I’ve gathered, aleph-O represents the natural numbers, and could be pictured as (X, X+1, X+2…) This can also represent the coordinate system for a multi-dimensional space. Aleph-1 can then represent the reals between 0 and 0.00000001(X<—>nonX) and still be considered larger than aleph-0. Aleph-2 seemed like a kind of “natural number” as it is the result of function that is applied to aleph-1. This comparison to natural numbers could be totally wrong. But it was confirmed for me by someone who was knowledgeable in the subject. And the comparison explains for me why it is said there are an infinite number of aleph values.

I did just find this, which is interesting, and I may look into it further or someone can explain it:

Aleph 1 is just the next largest infinity after the integers, which is not necessarily the reals. The statement that aleph 1 is the reals is the continuum hypothesis which can neither be proven nor disproven under the usual axioms of set theory.

Assuming the continuum hypothesis, aleph 1 is the reals, and assuming the generalized continuum hypothesis aleph 2 would be the set of subsets of reals (or the set of subsets of subsets of naturals). Then aleph 3 would be the set of subsets of subsets of reals, or the set of subsets of subsets of subsets of naturals, and so on.

Regardless of the status of the continuum hypothesis, aleph 1 is the cardinality of the set of countable ordinals. Then aleph 2 would be the cardinality of the set of at most aleph 1-sized ordinals, and so on.

Edit: The reference on the quotes in case someone thought they were generated by ChatGPT

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When I first heard about them the examples were…

aleph 0: the natural numbers
aleph 1: the real numbers
aleph 2: the number of possible functions on a real interval.

The truth however is that we have no real way of proving that these are examples of these things. We can only prove that the real numbers cannot be put in 1-1 correspondence with the natural numbers and this points to them being of a higher order of infinity.

The last seems fairly reasonable if we accept the expression…
aleph n+1 = 2 to the power aleph n
(which is an odd sort of formula if you look too closely at it)
…and we cannot prove that there are no orders of infinity between these and this is cause for some uncertainty regarding the above examples.

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https://www.nature.com/articles/d41586-023-00977-3

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The different answers are a clue to the real problem. ChatGPT isn’t designed to provide factual answers to factual questions. In the words of James McGrath, it’s not the computer from Star Trek:

OpenAI has emphasized from the outset what Chat-GPT is. It emulates human speech patterns. It does this remarkably well. They have been very clear what it is not. It is not a search engine. It is not designed, as it shuffles language into new patterns, to preserve and convey things that humans would identify as factual answers to our questions. It does not understand what you are saying nor what it is saying, and thus has no capacity to identify facts in the enormous sample of human text that serves as its basis for its own generated text.

You know how someone would address the computer on Star Trek and ask it for information? You can do that, and Chat-GPT will answer in something more like human speech than the computer on Star Trek. But because it does not provide information and has no capacity to do so other than accidentally on occasion as a byproduct of shuffling textual patterns that have information woven into them, it made things up.

McGrath wondered about John the Baptist’s statement that he wasn’t worthy to carry Jesus’ sandals, so the religion prof asked a few questions to ChatGPT:

Please list all references to someone carrying sandals in ancient Jewish literature.
Thank you, these are very helpful. Are there others outside the Bible?
Who might carry someone else’s sandals in the ancient Mediterranean world?

As you can see, even though I never asked about John the Baptist, Chat-GPT had in its textual corpus more mentions of John the Baptist together with strap and sandal than probably all other references combined. It told me what I already knew, that this is traditionally interpreted as a sign of humility. It invented a quote from the Dead Sea Scrolls and mentioned Geza Vermes, not because it is trying to pretend to know but because these kinds of words are in relevant text from which it shuffled and reorganized. It doesn’t know it is lying, even though it seems as though it is intentionally trying to make made-up information seem authentic. Can something that isn’t a self-aware person lie? But by virtue of how it designed, when it does what it was designed to, it ends up imitating human speech as though it were the speech of an untrustworthy person prone to fabrication. It weaves in details that make what it says sound plausible not because it has a desire to deceive (it has no desire whatsoever) but because details that are correct happen to show up as it creatively shuffles what others have put in text for with related key words.

So please, for the love of all that’s holy, stop quoting ChatGPT at us. I’d like to see the mods ban it, frankly.

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I keep forgetting you’re a math guy, as well as @mitchellmckain.

I thought this article was fascinating and it seems to answer the questions posed above. I’ll leave that judgment to the more mathematically educated, so please weigh in. If you have any interest in the history of mathematics, the article is definitely a must read. For the rest of us, I’ll post some TL/DR screenshots as a half-baked summary.

(Okay, in hindsight I went nuts on the screenshots, but the article’s a freebie so cut me some slack.)

https://www.cantorsparadise.com/the-nature-of-infinity-and-beyond-a05c146df02c

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One for @Mervin_Bitikofer

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Edit: Missed this one. Sorry, Merv

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The conclusion seems to refute this quote that Mike found:

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