Does the set of all sets have ontological value?

The set of sets only exists in our imaginations. All infinite sets are purely things of the imagination. They are a mental approximation of the very large that can never be realised:

1. Creating anything infinity large is impossible; not enough time / would never finish
2. Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller
3. Only in our minds can things continue ‘forever’; in reality this would be akin to magic

Quoting Devans99

1. Creating anything infinity large is impossible; not enough time / would never finish

Quoting Devans99

2. Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller

Quoting Devans99

3. Only in our minds can things continue ‘forever’; in reality this would be akin to magic

agreed, there is no actualized infinite. meaning that there is no set of all sets, or there is. as stated in the other thread...

Quoting TheGreatArcanum

The existence of a set of all sets is only a contradiction if the ground of all being, that is, the set of all sets in nature as opposed to in the imagination, does not reside in a paradox or a contradiction. if the set of all sets in nature has ontological value, and both contains itself and does not contain itself at the same time and in the same respect, there is no contradiction. it just means that the ground of being is paradoxical, which is the case. to say that there is no first set is even more absurd than saying that there is, for to say that there is no first set is to say that all being has its origin in absolute non-existence, or the absence of essnce altogether, or rather, the non-potential for existence to be, yet existence is, so absolute non-existence cannot be, meaning that existence must be contained within that which has an essence, and that which has an essence is necessarily a set. the set of all sets must exist.

Quoting TheGreatArcanum

agreed, there is no actualized infinite. meaning that there is no set of all sets, or there is.

It has to be one or the other:

- I think the set of all sets exists only in our minds along with all the other infinite sets.

- I think the set of all objectively real sets may exist, but it is not infinite.

From the other thread:

'on paper, yes, but it may be the case that something existent can both contain itself and not contain itself at the same time, in which case, it would not matter if it’s a contradiction or not.'

I don't think thats topologically possible.

Actually the set of all objectively real sets would have to be defined to exclude its own power set else we have problems.

I'm not sure the set of all sets can be defined properly without various restrictions...

Quoting Devans99

I don't think thats topologically possible

meditate on it a little more; what both exists, yet isn’t tangible, and both contains itself and does not contain itself simultaneously?

Quoting TheGreatArcanum

meditate on it a little more; what both exists, yet isn’t tangible, and both contains itself and does not contain itself simultaneously?

No idea.

Quoting Devans99

No idea

all things are born out of it, return to it, and exist within it at all times, but it was never born, and will never die.

Quoting TheGreatArcanum

that’s if you can prove that something cannot exist eternally, but to do that, you would have to prove that existence came into being out of non-existence, or rather, the non-potential for existence to be.

- Something cannot exist eternally in time because it would have no temporal start
- Eternal existence outside of time is however possible.

This argument from Aquinas sums it up:

1. Can’t get something from nothing
2. So something must have existed ‘always’.
3. IE if there was ever a state of nothingness, it would persist to today, so something has permanent existence.
4. It’s not possible to exist permanently in time (an infinite regress; it would have no start so could not be), so the ‘something’ must be a timeless first cause.

that’s if you can prove that something cannot exist eternally, but to do that, you would have to prove that existence came into being out of non-existence, or rather, the non-potential for existence to be.

Quoting TheGreatArcanum

does the set of all sets have an essence and therefore ontological value, or does it exist only in our imaginations?

Essence is simply a way of thinking about things--it's what an individual considers necessary features to apply a concept term as they've formulated the concept.

Quoting Terrapin Station

Essence is simply a way of thinking about things--it's what an individual considers necessary features to apply a concept term as they've formulated the concept.

Example:
An iron atom is not iron, but only a definition of the set of electrons, neutrons and protons together with their charge. So iron itself is a set of iron atoms, but the atoms aren't iron.

Reply to Christoffer

Right, so re your concept of "iron," more than one "iron atom" is necessary to have "iron" (without "atom" appended). So the essence of iron for you is that there's more than one atom of a particular type. (And maybe more than two . . . I don't know how many you'd require.)

This user has been deleted and all their posts removed.

Quoting tim wood

No. Iron is an element. One iron atom is iron. That's why it's called an element. Lots of things are not elements, like water. Water cannot exist at a level less than a molecule of water, a particular binding of hydrogen and oxygen. Or did you mean something else?

Even if an iron atom is an element, it's still a certain group of neutrons, protons and electrons which we designate the definition of iron. But I think you improved my point because it's better to make that point with molecules rather than atoms. One water molecule isn't water, it needs to be a group of molecules to be water in any term of the word we define it.

This user has been deleted and all their posts removed.

Quoting Terrapin Station

Essence is simply a way of thinking about things--it's what an individual considers necessary features to apply a concept term as they've formulated the concept.

essence is the aspect of a thing which remains unchanging so long as it exists, and as an abstract object in memory thereafter. quality is the aspect of a thing which remains constantly changing so long as it exists.

Quoting tim wood

If it exists in the imagination does it exist or does it not exist? Start with two: does two exist? And do ideas have no essence and no ontological value?

that depends on how you define “exist.” a mental image exists as an object of imagination, but not as an actualized physical object. the set of all sets is presupposed to have no ontological value, that no set of concepts or things which both does and does not contain itself exists.

Quoting TheGreatArcanum

essence is the aspect of a thing which remains unchanging so long as it exists,

No such thing.

Quoting Terrapin Station

No such thing.

here’s an experiment for you: go grab any object from the room you’re in and hold it in your hand, look at it, and then ask yourself, “what is this object?” a few seconds later, ask yourself the same question, and then again and again..repeat this experiment ten times; and if your answer doesn’t change, then you’ve just disproven yourself, no matter what you say, there is still some aspect which remains unchanged throughout, and that aspect points to the essence of that thing.

Quoting TheGreatArcanum

here’s an experiment for you: go grab any object from the room you’re in and hold it in your hand, look at it, and then ask yourself, “what is this object?” a few seconds later, ask yourself the same question, and then again and again..repeat this experiment ten times; and if your answer doesn’t change, then you’ve just disproven yourself,

The answer changes even though I say the so-called "same thing," because nothing is literally identical through time. The idea of something being the same through time is an abstraction--and abstraction that itself is different at different times.

What we answer--say that it's a bottle or whatever, is an abstraction that we've created. The object fits the concept we've constructed. Essences are the necessary aspects of our conception, what we require to call some x "a bottle" (or whatever the concept at hand).

I'm a nominalist, by the way.

Quoting Terrapin Station

The answer changes even though I say the so-called "same thing," because nothing is literally identical through time. The idea of something being the same through time is an abstraction--and abstraction that itself is different at different times.

What we answer--say that it's a bottle or whatever, is an abstraction that we've created. The object fits the concept we've constructed. Essences are the necessary aspects of our conception, what we require to call some x "a bottle" (or whatever the concept at hand).

I'm a nominalist, by the way.

the answer doesn’t change. yes, the qualities of that thing change over time, but the abstract set in which those changes occur within remains unchanged so long as it exists. hence the reason why an apple seed is not a pear seed, a ripe apple is an apple and not a pear, and a rotten apple is an rotten apple and not a rotten pear; no matter what state of being the apple is in, whether it is just a seed or decayed almost completely, all of those changes are still subsumed under its identity set, a purely abyss r set which must precede the existence of any changes which occur within it. if this set did not exist, apples could become inside of trees and not on twigs, or inside of the ground, etc...yet this is not possible, why? Because thins are first differentiated in the abstract before they come into being.

That abstract set doesn’t come into being after the object, and once we’ve conceived of it in our minds. that notion is completely absurd given the aforementioned reasons.

it doesn’t matter what your beliefs are, when provided with truths that contradict your opinion, you are expected to change your opinion to accommodate the truth, otherwise you are not worthy of the title or philosopher.

Quoting TheGreatArcanum

the answer doesn’t change.

Yes, it does, as nothing is identical through time.

Quoting Terrapin Station

as nothing is identical through time.

how about the concept ‘nothing is identical through time’? does that change over time? don’t you realize that you’re contradicting your own position as you speak of it?

Quoting TheGreatArcanum

how about the concept ‘nothing is identical through time’? does that change over time?

It's not as if this is hard to figure out. If nothing is identical through time, then "Nothing is identical through time" isn't identical through time.

You're conflating "not identical" and "isn't the case/isn't true" (on at least one instance).

Not at all the same idea.

Reply to Terrapin Station Quoting Terrapin Station

It's not as if this is hard to figure out. If nothing is identical through time, then "Nothing is identical through time" isn't identical through time.

You're conflating "not identical" and "isn't the case/isn't true."

Not at all the same idea.

this isn’t that hard to figure out, either the phrase ‘nothing is identical over time’ is identical to itself from one moment to the next, or it is not. if it isn’t identical to itself over time, well then what does it become? and if it is identical to itself, you’ve contradicted yourself. so what can the set of words and concepts ‘nothing is identical over time’ which to an abstract meaning, become, without the meaning being lost? According to your understanding, the meaning can and must be changed over time, so what does it become, if you don’t mind me asking? you seem to be avoiding what I’m saying and simply reverting back to your own position which has been shown to be contradictory. like I said, one cannot be a lover of wisdom and lover of lies at the same time.

Quoting TheGreatArcanum

either the phrase ‘nothing is identical over time’ is identical to itself from one moment to the next, or it is not. if it isn’t identical to itself over time, well then what does it become?

It's a non-identical "nothing is identical over time" at the different time.

You seem to be unfamiliar with nominalism, by the way.

Quoting tim wood

No. One water molecule is exactly water.

Can you add properties like viscosity, gas, solid state to one water molecule? Can you hold it, drink it? Is the air essentially water since we have humidity levels? We say different things about water based on its form as a group of water molecules, but we cannot add any properties whatsoever to only one water molecule. Therefore, that water molecule is not specifically water, it is distinctly a water molecule. The essence of water, with all the properties of water as a group of water molecules, is out of them being a group, not a single molecule.

This user has been deleted and all their posts removed.

Reply to tim wood

So your counter argument now is that I should give up? Did you try understanding what I'm aiming for here? What I wrote also applies to an iron atom. I think you are mistaking the science with what I refer to, which is how we define material in language. No iron atom or water molecule works as "iron" or "water" in any terms of how we define the properties of them. If you deal with this discussion as a chess match of who makes most "mistakes" as you put it, then I'm not interested, and that is not what a philosophical dialectic is about.

if the essence of a thing involves only that unseen aspect which remains unchanged so long as it exists, how is that we can have knowledge of the essences of things even despite the fact that we cannot see their unchanging microcosmic structures like 'iron atoms' or a 'collection of iron atoms'? the substance that its made of must be combined with some abstract form for the essence of an object, say, like a hammer, to be known, yes? Where lies the origin of that abstract form? Is that origin different for a natural object as opposed to an object made by man, which is in some sense 'natural' because man is entirely a natural object?

This user has been deleted and all their posts removed.

If comedy has ontological value and

if you like the tongue twister

how many sets could the set of all sets set if the set of all sets set sets

then it holds without possible objection the set of all sets has ontological value

Reply to TheGreatArcanum A set is a selection of items with specific attributes. To talk of a set of things with attributes is to exclude nothing.

This is no more than a misapplication of linguistics to abstract logical concepts.

Sets only have meaning in relation to other possible sets. To talk of ALL sets is nonsense as it is to talk of ‘backwards yellow’ or ‘big shaped flavours’ - such strings of words are of use in a playful artistic endeavor.

Quoting I like sushi

A set is a selection of items with specific attributes. To talk of a set of things with attributes is to exclude nothing.

Yes, a set is a defined selection of specific qualities that are distinguished from other sets and also from nothingness, that is, an empty set with no qualities.

Quoting I like sushi

This is no more than a misapplication of linguistics to abstract logical concepts.

Quoting I like sushi

Sets only have meaning in relation to other possible sets. To talk of ALL sets is nonsense as it is to talk of ‘backwards yellow’ or ‘big shaped flavours’ - such strings of words are of use in a playful artistic endeavor.

sets have meaning only in relation to other sets, OR in relation to themselves. if a set exists, it persists, and if the qualities of a set are changing and the set is therefore expanding, which is true of sets that are ontological and not imaginary, that set is identical to itself from moment to moment in time in its essence, but not identical to itself from moment to moment in time in its quality.

now, ontologically speaking, this set is either contained within itself, or it is contained within a higher set. if it is contained within itself, there is no higher set, meaning that it is the set of all sets, if it is not contained within itself, it is contained within a higher set, and this chain either goes on to infinity and an infinite regress ensues, or it does not, in which case it ends in a set of all sets. it’s very simple.

Reply to TheGreatArcanum That’s gibberish. You cannot agree AND disagree with what I said. You admit it is a linguistic issue then persist that it isn’t.

You cannot have it both ways friend.

Consider the game of chess. There are set rules. Of you break the rules you can insist that you’re still playing chess, but I’d disagree for obvious reasons. This is no different. A set is a set, there is no “rule of rules” other than in your imagination.

A set cannot be related to itself. That is plainly contrary. You see to think it reasonable to extend abstract thought into physical reality without even batting an eyelid? Maybe because your eyes are already tightly shut? ;)

You say this (which is utter contrary gibberish):

now, ontologically speaking, this set is either contained within itself, or it is contained within a higher set. if it is contained within itself, there is no higher set, meaning that it is the set of all sets, if it is not contained within itself, it is contained within a higher set, and this chain either goes on to infinity and an infinite regress ensues, or it does not, in which case it ends in a set of all sets. it’s very simple.

I agree. It is simple. It is simply gibberish.

Note: you may think this means something and you may well actually have a point to make that is buried in your head. You’re failing to express whatever it is OR the point you wish to make is so deeply flawed you’re just going around in circles. That is my honest view of what you’ve shown so far.

Quoting tim wood

if one atom of gold is not gold, then your ring, perhaps billions of atoms, cannot be gold, because no part of it is gold

That's just my point; the characteristics of gold is only through a set of atoms, not one single atom. Even in gas form, it's the group of atoms which makes up gold, a single atom is neither gas, solid or liquid gold, it's specifically a gold atom. So the essence of gold is through the group of gold atoms making up gold, not single atoms. Point being, how we define the materials around us aren't based on the singular atoms, but the grouping of those atoms. A ring cannot be made of one gold atom, only a group of atoms, therefore, how we work and view gold as a substance does not come from the singular atom, but the group that makes up how we define it as a substance. Even in chemical reactions, one single gold atom is impossible to make enough reactions when in contact with a group of others. You wouldn't say that an iron dagger which has one gold atom in it, is an iron-gold alloy.

It's not really about what is technically true here, it's about language, how we define things. Technically, one gold atom is gold by the makeup of its neutrons, protons, and electrons, but no one would define gold as a substance with the essence of gold as we view it, if there was only one atom. Because the properties of gold as we know them comes from a group of those atoms, not one atom.

Reply to I like sushi according to your understanding, it seems, the abstract set which contains all the parts of a thing, and distinguishes it from other things, comes into being when we conceive of it in our minds, while my understanding is that the abstract set precedes our conception of it, and also, preceded the existence of the object itself. According to my understanding, the abstract precedes the concrete, the concrete is contained within the abstract, and abstract thought precedes concrete things which are, in their truest essence, abstract ideas that have become actualized into existence by means of subjectivity relative to an a priori limited sense perception.

think about what it means for you to say that the law of identity, an abstract concept, doesn’t come into being until after man conceived of it, or rather, that the law of identity is a subset of man and man, and nature itself, is not a subset of the law identity. this is what your position asserts. think about what that means, it means that existence can be become non-existence from one moment to the next in time, or be equal to non-existence. it’s quite literally the most absurd position ever held, ever, at any time or anywhere.

Reply to TheGreatArcanum I think I get the gist of what you’re attempting. The problem is you’re necessarily bound to the use of language (as in the words you’ve written) and if this isn’t addressed it looks more like mysticism dressed up as logic.

“I” exists because if language. Without language there is no “I”. That is to paraphrase part of an argument I heard some time ago form who I don’t recall? The point being the meaning of “I” is bound in language. To talk of something ‘outside’ of language is to pull such a “thing” into the sphere of language thus destroying its non-language attribute. In this sense language is a means of adumbrating what isn’t language.

So I have issue with what you’re expressing in regards to how you’re dealing with it ontologically and epistemologically - each being necessarily parts of each other.

There is no doubt this is a seriously tricky topic because we’re stretching the use of language (and I am suggesting you’re overreaching).

Reply to I like sushi language isn’t defined by human language. logic is language, mathematics is language, so of course, what am saying is bound by language, yes, but not human language. man didn’t invent logic or mathematics, he discovered it.

the ‘i’ doesn’t necessitate language, only a direct apprehension or intuition of the will as a causal entity, that’s how one knows that they have an ‘i’ and this precedes language.

I am not overreaching. Once you become a mystic you have no other choice but to support a mystical philosophy, and this is an understatement. I speak the truth, read more of my philosophy and you will understand clearly what I mean and why it’s absolutely true.

Reply to TheGreatArcanum

I am not overreaching. Once you become a mystic you have no other choice but to support a mystical philosophy, and this is an understatement. I speak the truth, read more of my philosophy and you will understand clearly what I mean and why it’s absolutely true.

Does that mean you admit it is mysticism?

the ‘i’ doesn’t necessitate language, only a direct apprehension or intuition of the will as a causal entity, that’s how one knows that they have an ‘i’ and this precedes language.

Wrong. That is not what was meant. You’re referring to an item preceding language (or rather you THINK you are) with no logical justification.

Logic without a medium is NOT logic. If logic is language then why are there two word concepts? Are they the same thing? If not what is the difference?

Quoting I like sushi

Does that mean you admit it is mysticism?

yes, if you still haven’t achieved the mystical union, you’ve yet to achieve the pinnacle of human evolution. as a result, you remain, in comparison to what you could be, primitive and unevolved

Quoting I like sushi

Wrong. That is not what was meant. You’re referring to an item preceding language (or rather you THINK you are) with no logical justification.

go into meditation. move your awareness from its natural center to the tip of your finger and back to its center. now, do this again in various places around your body. you’ve just disproven your theory that one’s cognition of their own existence requires language. it really only requires will and imagination in combination with memory.

the only medium logic needs is memory and intuition, as well as will and imagination. this is empirically verifiable within oneself. one doesn’t need to speak or think the words “I exist” to know intuitively that they exist; to say so is beyond absurd.

Reply to TheGreatArcanum Never mind. Have fun in your mystical little garden :)

I'm going to be the surrogate forum moderator here, and am going to be flagging ad hominems that arise.

More chimp-pig content, please.

Reply to I like sushi Reply to TheGreatArcanum

Stahp.

@TheGreatArcanum, it's puzzling that you don't subscribe to Platonism and assert that the set of all sets has ontological value. Can you lay out your reasoning here?

Quoting Wallows

it's puzzling that you don't subscribe to Platonism and assert that the set of all sets has ontological value. Can you lay out your reasoning here?

the set of all sets is the set of all memories, and Memory, that is, a perfect, infallible, Absolute Memory, just the same as the set of all sets, both contains itself and does not simultaneously. This Absolute Memory is identical with the Law of Identity itself, meaning that each change occurring within the Absolute Memory is equal to itself so long as it exists, and is stored as it is in relation to all other changes in the Absolute Memory. I say that the set of all sets has "ontological value" because there is some thing which is both completely abstract and existent that both contains itself and does not simultaneously. This is the greatest discovery that any philosopher has ever made. That philosopher is me.

Psychoceramics...

Reply to Wallows you wouldn't think so if you saw my argument, which is as simple as it is elegant.

Quoting TheGreatArcanum

you wouldn't think so if you saw my argument, which is as simple as it is elegant.

You might as well say that God is the set of all sets that is epistemically closed in a solipsistic manner. I digress.

the essence of existence must be preserved over time because that which exists, persists in existing. the essence of a thing is abstract. the only possible thing that can preserve the essence of an abstract concept over time is memory.

Quoting Wallows

You might as well say that God is the set of all sets that is epistemically closed in a solipsistic manner. I digress.

Well, I have a definition for God, of course. If God exists, God cannot be not Self-Aware, that which is self-aware necessarily possesess a will, an imagination, and a memory; so to prove the existence of God, one need only prove that the set of all sets involves memory in its essence, for where there is memory, there are abstract concepts, where there are an expanding number of abstract concepts, there is imagination, and where there is imagination, there is willing, that is, subjectivity.

Reply to TheGreatArcanum

Imagination, willing, and subjectivity, and God... It's hopeless to try and draw out how you see any coherence between these terms used.

But, anyway, given that the set of all sets is epistemically closed off from any other set, then it is "absolute objectivity" to borrow your phrase. So, "absolute objectivity" to beat the phrase is in essence, God manifest. Yeah?

Quoting Wallows

Imagination, willing, and subjectivity... It's hopeless to try and draw out how you see any coherence between these terms used.

But, anyway, given that the set of all sets is epistemically closed off from any other set, then it is "absolute objectivity" to borrow your phrase. So, "absolute objectivity" to beat the phrase is in essence, God manifest. Yeah?

its not hopeless to draw out why imagination, willing, and memory which together as a trinitarian unity define subjectivity, because where there is one there is necessarily all three. This is because the essence of each necessarily involves the other two, one cannot will without memory or imagination, for example...


it is closed off, so to speak, from all of its past memories, but not epistemologically closed off in the sense that it can retrieve them in its present awareness by means of will and imagination. Absolute Objectivity involves only those aspects of being which remain eternally unchanged, that is, Absolute Memory, or the Absolute Law of Identity (E(t?) = E(t?)) and the Absolute Law of Non-Contradiction (E (t, infinity) ? ?E (t, infinity)), that is, Absolute Time or Duration, which together formulate "Absolute Objectivity" i.e. (E (t, infinity) = E (t, infinity) ? ?E (t, infinity)). Absolute Objectivity is God Unmanifest. Manifest God doesn't spring forth from the Unmanifest until the Law of Excluded Middle (E (t1, Will) v ?E (t1, Will)) comes into being (it may be eternal, it may not be; I haven't decided if it is, or if I can know whether it is or not yet, I just know that it exists). and from there, there is a dialectical developmental process of Consciousness, in which, the change created by the will creates a new concept in memory which is then preserved perfectly as it was in relation to all other current states of being, eternally thereafter, and as a result, the Absolute level of Intuition and Self-Knowledge is raised.

Reply to TheGreatArcanum

I sense Hegel here. Is that where you're deriving your rationale here because it needs some grounding I suppose.

Reply to Wallows No, Hegel is a fool, his dialectical process is pretty much correct, but his philosophy is so convoluted and poorly written that he did more harm to idealism than he did good. My philosophy is a mix between Schopenhauer's, Hegel's, Descartes', Spinoza's, and Leibniz's. Believe me when I say that I've expounded on these concepts a great deal and my phenomenology of the will alone is going to make me famous.

I digress.

Quoting Wallows

I digress.

if the law of identity isn't eternal, then it came into being once upon a time. if it is eternal, everything that exists is contained within an eternally abstract concept, meaning that space is an illusion and only time and consciousness are absolutely real. if you want to make the case that the law of identity came into being once upon a time, by all means, go right ahead. you can't explain that away without contradicting yourself. hence the reason philosophers today, who are more so fools than they are wise men, avoid the nature of the ground of being altogether and then proceed to create philosophies without knowing its nature. philosophy isn't dead, only man's intellect is dead. All of these post-modern philosophers are, in the universal sense of the word, idiots.

I've not read much of the above. I merely comment that I seem to remember that Russell's Paradox was dismissed by Wittgenstein as being 'aberrant language'. IMO the substantive issue is breakdown of static classical set theory in the context of dynamic perceptual states. If, for example, we consider 'fuzzy sets' in which 'the law of the excluded middle' does not apply then the so called 'paradox' disappears because 'set membership' is fluid.
A similar issue seems to be the suggested of Cohen's award of the Field's Medal, for proving both that there was and was not 'another infinite set of cardinality between Cantor's infinite sets.
Classical logicians beware ! :cool:

Quoting fresco

A similar issue seems to be the suggested of Cohen's award of the Field's Medal, for proving both that there was and was not 'another infinite set of cardinality between Cantor's infinite sets.

Good God. Cohen did nothing of the sort. He showed (in conjunction with ?Gödel) that CH was formally independent of ZFC. ?Both ?Gödel and Cohen believed that CH is false -- in other words, that it has a definite truth value. Just one that's not accessible via ZFC.

You are confusing syntax with semantics, formal systems with models.

By the way you even stated CH incorrectly. CH doesn't say that there's a set "between Cantor's infinite sets." Rather, the negation of CH is that the real numbers have a cardinality that's larger than Aleph-1. ?Gödel believed the reals had cardinality Aleph-2. Cohen thought it might be much larger than that. But all the Alephs are Cantor's cardinals.

Thankyou. I stand corrected on the technicalities of Cohen's work. But as an example of 'problems' with classical logic I still claim validity.
I have no idea where you are hoping to go with my alleged 'confusion' between syntax and semantics etc. As far as I'm concerned the contexts in which you want to differentiate between those terms is nothing to do with the context of my anti-classical logic position.
'

Quoting fresco

Thakyou. I stand corrected on the technicalities of Cohen's work. But as an example of 'problems' with classical logic I still claim validity.
I have no idea where you are hoping to go with my alleged 'confusion' between syntax and semantics etc. As far as I'm concerned the contexts in which you want to differentiate between those terms is nothing to do with the context of my anti-classical logic position.

I only read your post and commented on your remarks regarding CH. I didn't take a position on logic. Sorry for any confusion.

The point about syntax and semantics is that in terms of syntax, we can neither prove nor disprove CH within ZFC. But we can exhibit a model, or interpretation of ZFC, in which CH is true (?Gödel 1940) and another model in which it's false (Cohen 1963). In any given model of ZFC, CH has a definite truth value. It's either true or false. That's semantics. But syntactically, we have no proof.

As I say if you are making a larger point, I didn't address it.

Quoting fresco

Russell's Paradox was dismissed by Wittgenstein as being 'aberrant language'.

This is an appeal to authority fallacy; and a poor appeal at that, for no matter what language we decide to use to refer to the ground of existence, that is, the origin, container, and final destination of all words, concepts, objects, and motions, it remains ontological and metaphysical.

Its a bit ironic that the 'authority' to which you object was the one who identified the 'language on holiday' you produce in your objection.. Only you know what you are talking about when you speak of your quasi-religious 'final destination'.. If you know the apocryphal tale of "turtles all the way down" it seems to apply to you.

[i]The linguist John R. Ross also associates James with the phrase:

The following anecdote is told of William James. [...] After a lecture on cosmology and the structure of the solar system, James was accosted by a little old lady.
"Your theory that the sun is the centre of the solar system, and the earth is a ball which rotates around it has a very convincing ring to it, Mr. James, but it's wrong. I've got a better theory," said the little old lady.
"And what is that, madam?" inquired James politely.
"That we live on a crust of earth which is on the back of a giant turtle."
Not wishing to demolish this absurd little theory by bringing to bear the masses of scientific evidence he had at his command, James decided to gently dissuade his opponent by making her see some of the inadequacies of her position.
"If your theory is correct, madam," he asked, "what does this turtle stand on?"
"You're a very clever man, Mr. James, and that's a very good question," replied the little old lady, "but I have an answer to it. And it's this: The first turtle stands on the back of a second, far larger, turtle, who stands directly under him."
"But what does this second turtle stand on?" persisted James patiently.
To this, the little old lady crowed triumphantly,
"It's no use, Mr. James — it's turtles all the way down."

—?J. R. Ross, Constraints on Variables in Syntax 1967[10][/i]

Reply to fishfry Reply to fishfry
"CH has a definite truth value. It's either true or false. That's semantics. But syntactically, we have no proof".
I'm not clear what you mean by 'syntax' here. The 'semantic point' is that the phrase 'definite truth value' automatically invokes the semantic context of classical binary logic.

....on further consideration, I assume you mean 'rules governing what constitutes a valid form of answer'. On that assumption we are touching on 'Zen Koan' territory which forces the pupil to consider the assumptions regarding the structure of 'the question'.. In that case my identification the inapplicability of the rules behind the assumptions of classical logic could be regarded as a 'syntactic' point

Quoting fresco

"CH has a definite truth value. It's either true or false.

That is a Platonic claim. It can be strongly argued against. I"m not taking a position one way or another but only pointing out that your claim is arguable.

Consider a variant of the game of chess in which pawns may be promoted to queens or rooks but not knights or bishops. That is not a very radical change in the rules. There are in fact many variants of chess.

Now we come upon two expert chess players arguing over which version is true. But we can see that there is no truth of the matter at all. Chess and variant-chess are formal games. We make up the rules arbitrarily. The only requirement is that the rules are sensible enough so that the game is playable; and that enough people find it fun and enjoyable to play. There is no requirement with truth.

To a formalist, math is the same. It's a meaningless game played with marks on paper according to rules.

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

To a formalist, CH has no definite truth value. We can play the game with CH or with its negation. And when it comes to CH it's a very interesting situation. All of the new axioms which set theorists have studied in order to get a handle on CH are CH-agnostic. You throw in a new large cardinal axiom, for example, and there's a CH and a not-CH version.

Now if someday some physicist determines that ZFC is instantiated in the physical world, then CH would become a research project and would have a definite truth value.

Till that day, if it ever comes, we can only ask if CH is true in the "correct true model of set theory out there somewhere." And the very existence of such a world is a Platonist dream. Gödel himself, as I've mentioned, was a Platonist. His incompleteness theorems to him mean that there is a realm of mathematical truth that's not accessible to the axiomatic method of symbol manipulation.

All that is by way of saying that when you say there is a definite truth value to CH, you might as well ask how pawns may "really" be promoted. The question is a category error. There is no truth in formal games.



Quoting fresco

That's semantics. But syntactically, we have no proof".
I'm not clear what you mean by 'syntax' here. The 'semantic point' is that the phrase 'definite truth value' automatically invokes the semantic context of classical binary logic.

Our syntax consists of:

* An alphabet of symbols;
* The usual rules by which we can form well-formed logical and mathematical formulas;
* The inference rules of first-order predicate logic, by which we can start from a set of wffs called the axioms, and derive other wffs called the theorems. Note by the way that an axiom is a theorem, since every axiom A has a one-line proof, namely A.
* The axioms of ZFC.

I should mention that the rules for wffs and the rules of inference are computable. You could write a program (ie Turing machine) to recognize a valid wff and a valid inference.

It is a fact that there is no proof in ZFC of CH nor its negation. That's syntax.

Semantics is an interpretation. Some universe of set theory, called a model, in which CH or not-CH are a matter of observable fact. The question is whether there is a Platonic "correct" model of set theory that settles the issue of CH. A lot of smart people haven't found one yet.


Quoting fresco

....on further consideration, I assume you mean 'rules governing what constitutes a valid form of answer'. On that assumption we are touching on 'Zen Koan' territory which forces the pupil to consider the assumptions regarding the structure of 'the question'.. In that case my identification the inapplicability of the rules behind the assumptions of classical logic could be regarded as a 'syntactic' point

No, nothing so woo-woo. A simple matter that syntax, the formal rules of deriving theorems from axioms, does not settle the question of CH when starting from ZFC. One can then find interpretations of the symbols in which CH is objectively true; and interpretations in which it's objectively false. And nobody knows an interpretation of set theory that's so obviously "the right one" that we're willing to call it the official model and thereby determine the truth value of CH.

I hope this wasn't too wordy and addressed some of your concerns. Syntax = derivations, semantics = interpretations.

Reply to fishfry
?
The original quote about 'definite truth value' was yours not mine.

My reconsideration of 'syntax was based on my understanding of 'syntax' as the linguistic one of 'rules governing combination of components', sometimes called 'grammar'.
Thus
Does the dog bite the man?... has the same syntactic structure as... Does the man bite the dog?
The common syntax implies a yes/no answer, but the particular answer is based on semantics.

Quoting fresco

The original quote about 'definite truth value' was yours not mine.

You have a cat? Mine sometimes walks on my keyboard and writes half the stuff I post here.

Quoting fresco

"CH has a definite truth value. It's either true or false.

So you didn't write that? Ok.

No. I quoted you !
Maybe you quoted someone above.

Quoting fresco

Maybe you quoted someone above.

My cat did it.