It has a few problems. The biggest are 1) the proof uses two numbers to suggest that one of them is "foundational", and 2) the statement "All computers can perform any computation using two numbers" does not imply that computers must use two numbers.

Friendly technical note: axioms aren't proved, they're assumed. Using a set of given axioms, we prove theorems. Once we have a few theorems, we can use them to make other theorems. The set of all theorems is a theory.

Humor me to lay out a simple example, so you can get a better feel for what these things look like. We'll use group theory, which is one of the simplest algebraic structures (few axioms) that nevertheless is very rich (in theorems).

A group is just a set of elements \( S \) and a single binary operator \( \star \). The elements can be anything -- numbers, geometric shapes, colors, types of monkeys -- and the operator can be anything that obeys the group axioms, which are the same for every group.

Suppose a, b, and c are elements in \(S\). Then, the group axioms are:

• \( S \) is closed under the operation: \( a \star b \in S \)
• There is an identity element \( e \), such that \( a \star e = a \)
• The operator is associative: \( a \star (b \star c) = (a \star b) \star c \)
• Every element has an inverse under the operation: \( a \star a^{-1} = e \)

That's it. We don't have to prove any of the axioms, we just assume them. Using these axioms, we can prove theorems.

For instance, we know that the following is a theorem of any group:

If \( a \star a = a \), then \( a \) is the identity element of the group.

Proof:

1. \( a \star a = a \)
2. \( a \star a = a \star e \) (by identity axiom)
3. \( a \star a = a \star (a \star a^{-1}) \) (by inverse axiom)
4. \( a \star a = (a \star a) \star a^{-1} \) (by associativity axiom)
5. \( a \star a = a \star a^{-1} \) (by step 1)
6. \( a \star a = e \) (by inverse axiom)

QED. We've proven the theorem. This means that, for any group we can think of, we know that \( a \star a = a\) implies that \( a \) is the identity element. For example, the integers under addition form a group, \((\mathbb{Z}, +)\). From elementary school, we know that 0 is the identity element of this group. Indeed, 0 is the only element of \( \mathbb{Z} \) for which \(a + a = a\).

What about the integers under multiplication? Is \((\mathbb{Z}, \times)\) a group? It is not, because axiom 4 (inverses) is not obeyed for every element. For example, the number 2 has no multiplicative inverse in the integers.

Group theory, being so simple, is quite general. We can add more structure -- and, hence, more power -- by adding stuff to it. We can introduce new axioms (e.g., adding the commutative property leads to the theory of Abelian groups), or, even better, introduce more operators, which leads to richer structures, such as the rings and fields we learn about in elementary school arithmetic. Using these structures, we can create even larger palaces, such as vector spaces, Banach spaces, Hilbert spaces, etc.

This is generally how theories build on one another.

That’s very helpful, thank you. However, one of the reasons we’re working together is so you can do that sh*t.


You know what I’m tring to say above, right?

They “can” means they “don’t need more than that.” They are the H and the O of H2O and every other molecule involving them.

How about:

The number 1 ("one") and 0 (“zero”), are the foundational numbers from which all other numbers are conceptualized. All bases and computations can be done using just these 2 numbers.

Theorem: All modern computers are able to perform any computation of any kind using the number 1 (unary) or 1 and 0 (binary) as represented by classical bits, and yield results in any base. QED.

 

Using states of RAM, the computer I described has a (very simple) concept of "heat affects my performance". We can imagine this network of state associations becoming more complex by adding more types of sensors, more code, and more state, giving the computer a concept of its own general health.

Contrast this with another type of concept. Again, using states of RAM affected by different types of sensors and code, the computer can hold a concept of the current weather. To the computer, these two concepts are indeed distinct, which we can deduce by noticing that it responds differently to increases in internal vs external temperature. (In the former case, it might increase its fan speed; in the latter, it might suggest a day at the beach.)

That is very logical. The issue I see is, this means you are just a concept? And where specifically is the concept of "YOU" that seems to differentiate from all the other concepts?

And why is there a ready distinction in the BIOS that insists that "CONCEPTS" aren't necessarily "_REAL?" In fact, "_REAL" could be defined on some level as a concept's opposite, no? E.g., You had a concept of a dream wife. You met her. Your _REAL wife is the same or different from your _CONCEPT wife.

 

This potentially needs academic polish, but see how grokable it is as starting point:

Lots of problems, but we'll go through them.

Axiom 1: A number is a concept with a name.

Fine with this.

Axiom 2: A number can be grouped together in order to “number” or "count."

Weird use of "grouped" on a singular item, circular use of "number". This needs a lot of work.

In the most elementary of ways possible (unary), this means to add successive copies of the number 1 to create different numbers that can be given a unique name. They can also be labeled as a shortcut symbol. For example, “1111” in unary is “4” in base-10.

Fine, though a couple of technical notes. Try not to use "add" in a mathematical context unless you mean summation. Instead of "add successive copies", something like "concatenate successive copies".

Also, be mindful of the difference between a number and a numeral (symbol). Using the proper terminology, we'd say "successive copies of the numeral 1 to represent different numbers".

Also, be mindful of the differences between "unary" and "base-1". A unary system has one symbol, so base-1 is indeed a unary system, though unary is more general than numbers. The super precise way of saying what you said is, "For example, '1111' in base-1 is '4' in base-10."

Axiom 3: Beyond counting, one can create special computational numeric terms that are the result of appending a separate component of arithmetical operation(s).

This is more of an explanation than an axiom. I'm fine with it as explanatory text.

These are no longer numbers, but higher abstractive expressions that involve more numbers with analysis to cognize beyond simple counting.

This is a key point in your thesis. In a formal treatment, you'd want this to be a theorem: "computational numeric terms, as described above, are not numbers."

A number cannot be divided without creating a separate system to denote an abstract rule system for fractionation beyond counting.

Feels arbitrary to pick on division but allow other arithmetic operations. Subtraction leads to the integers, so you'll need to justify why division leading to the rationals is not "fundamental".

I get your point regarding numbers that are defined by arithmetic operations, and it has merit. It seems clear that the counting numbers (ℕ) are distinguished in that they're not born from arithmetic, just counting, so I'm totally with you on recognizing this important property.

But where I get iffy is in your claim that "arithmetic numbers" are not numbers. It seems to me that as fundamental as counting is, comparing magnitudes -- e.g., dividing -- is just as fundamental. If I have two sticks, I can compare their lengths and say "this one is bigger". Numbers quantify that notion of "bigger", with ratios -- fractional numbers -- quantifying the magnitudes.

For hundreds of thousands of years, humans walked around knowing which stick was bigger. They even had a system of ℚ to do comparisons: they would carve evenly-spaced notches on a stick of reference length, a caveman ruler. Using the counting numbers and the ruler, a caveman could say that the length of this rock is 3 notches on the stick. If the stick had 10 notches, then the rock is 3/10ths of the stick. The neighbor's rock is only 2/10ths of the stick; our caveman has the bigger rock.

You'll need to better justify how counting is more fundamental than magnitude comparisons.

Another example is what is commonly labeled as “Root 2” or “ √2.” This again is not a number, but an operation upon a number. Here we have 2 (or 11 in unary) as a number, with a “radical operation symbol” attached to it. The symbol is calling for a non-existent number that when multiplied by itself yields the number 2. This number is non-existent, so we call it “irrational,” which can also mean “unknowable.” In practical terms, this also means non-computable, uncountable as-is and therefore should not be treated as a number, or thought of as comparable to a number. Irrational numbers are the result of dividing two numbers that don’t rationally divide, and so an infinitesimal suffix is created that approaches another countable, rational number but doesn’t ever do so. It should therefore be called a “numeric expression” only.

I'm definitely not sold on this. For one thing, "irrational" is not used as a synonym for "unknowable", it literally means "not ratio-able", i.e., cannot be expressed as a ratio. And though this is a tougher conceptual pill to swallow (ergo Pythagoras), we cannot deny that there are magnitudes that cannot be expressed as the ratio of two counting numbers.

Using our caveman ruler example from above, suppose that the finest caveman carpenter used the straightest stick and carved the most finely-spaced notches. If the finest caveman artist drew a square in the sand using the stick and carefully measured the diagonal, they would find that -- no matter how fine the notches were spaced -- the length of the diagonal would never touch a notch.

Yet, no one would disagree that the diagonal has a length. So, why should the magnitude of this length not be a number?

“Transcendental numbers” are also not numbers, but expressions of multiple numbers written as single symbols. Pi/π is the ratio of two numbers. e is the base of the natural log.

This needs a lot of work. Pi is not the ratio of two numbers (which, to you, are only integers). In any case, Pi (and any other transcendental number) can also be a magnitude -- indeed, 180° is precisely equal to Pi radians.

Theorem: Based on the axioms above, |ℕ| < |ℝ| is an irrational comparison akin to saying "cat" is less than "dog." The first set is a set of numbers that only have 1 level of addition to create them (counting). The second set is a set of numeric computational expressions or computational results of variable complexity combined with countable numbers to create a set that shouldn’t exist as a true “numeric set” but “an abstract collection of numeric expressions.”

One cannot logically compare either set and say the cardinality of one is greater or less than the other’s due to this fundamental definition disparity.

This is patently false. Let A be a set of three dogs and B a set of five cats. Then |A| < |B| is a perfectly valid (and true) statement. The cardinality of a set is the count of its members, which is a unit-less number. Therefore, the cardinalities of any sets can always be compared.

 

There’s an implied definition there, though... “real” can mean here, “I exist as the very incarnation of MC, bruh, and you or anyone else don’t cut it.” He would then proceed to rattle off elements of the real set MC vs. poser set MC. “Check my rhymes, my ride, my flow, my dope, my girls, my backups, my Lex, my hoes; then tell me I ain’t the REAL DEAL, fulfilling the definition of MC Hilbert in physical space, in light of the definitions in your mind.”

LOL! You are the real MC.

The concept dog is “real” to you, as is the dog in physical space. Which one is “more real?” Are we going to ignore the “gradation element?”

That's exactly my point, it's absurd to consider something "more real" than another. When you can give me an example of something that is not "real", then I'll concede to using "real" as a meaningful word.

 

They “can” means they “don’t need more than that.” They are the H and the O of H2O and every other molecule involving them.

The difference is that H2O must have H and O, whereas computation need not be binary. Difference between necessary and sufficient.

The number 1 ("one") and 0 (“zero”), are the foundational numbers from which all other numbers are conceptualized. All bases and computations can be done using just these 2 numbers. are the foundational numbers from which all other numbers are conceptualized. All bases and computations can be done using just these 2 numbers.

The last sentence is fine, the rest are unjustified. We can write base-2 numbers in base-1 (or base-42). Therefore, base-2 is neither special nor privileged.

Theorem: All modern computers are able to perform any computation of any kind using the number 1 (unary) or 1 and 0 (binary) as represented by classical bits, and yield results in any base. QED.

I'm fine with this, but it's what's technically known as a "weak statement": it gives a sufficient condition for computation. A strong statement demands necessary conditions, e.g., "There are computations that can only be done with a binary system." The latter statement, though strong, is provably false.

 

That is very logical. The issue I see is, this means you are just a concept? And where specifically is the concept of "YOU" that seems to differentiate from all the other concepts?

Yes, yes, yes! The concept of SELF seems to be the defining characteristic of consciousness. Like all other concepts, the concept of SELF is a bunch of state associations. It's not located in one place.

And why is there a ready distinction in the BIOS that insists that "CONCEPTS" aren't necessarily "_REAL?" In fact, "_REAL" could be defined on some level as a concept's opposite, no? E.g., You had a concept of a dream wife. You met her. Your _REAL wife is the same or different from your _CONCEPT wife.

No idea what "_REAL" means.

 

Lots of problems, but we'll go through them.


Fine with this.


Weird use of "grouped" on a singular item, circular use of "number". This needs a lot of work.


Fine, though a couple of technical notes. Try not to use "add" in a mathematical context unless you mean summation. Instead of "add successive copies", something like "concatenate successive copies".

Also, be mindful of the difference between a number and a numeral (symbol). Using the proper terminology, we'd say "successive copies of the numeral 1 to represent different numbers".

Also, be mindful of the differences between "unary" and "base-1". A unary system has one symbol, so base-1 is indeed a unary system, though unary is more general than numbers. The super precise way of saying what you said is, "For example, '1111' in base-1 is '4' in base-10."


This is more of an explanation than an axiom. I'm fine with it as explanatory text.


This is a key point in your thesis. In a formal treatment, you'd want this to be a theorem: "computational numeric terms, as described above, are not numbers."


Feels arbitrary to pick on division but allow other arithmetic operations. Subtraction leads to the integers, so you'll need to justify why division leading to the rationals is not "fundamental".

I get your point regarding numbers that are defined by arithmetic operations, and it has merit. It seems clear that the counting numbers (ℕ) are distinguished in that they're not born from arithmetic, just counting, so I'm totally with you on recognizing this important property.

But where I get iffy is in your claim that "arithmetic numbers" are not numbers. It seems to me that as fundamental as counting is, comparing magnitudes -- e.g., dividing -- is just as fundamental. If I have two sticks, I can compare their lengths and say "this one is bigger". Numbers quantify that notion of "bigger", with ratios -- fractional numbers -- quantifying the magnitudes.

For hundreds of thousands of years, humans walked around knowing which stick was bigger. They even had a system of ℚ to do comparisons: they would carve evenly-spaced notches on a stick of reference length, a caveman ruler. Using the counting numbers and the ruler, a caveman could say that the length of this rock is 3 notches on the stick. If the stick had 10 notches, then the rock is 3/10ths of the stick. The neighbor's rock is only 2/10ths of the stick; our caveman has the bigger rock.

You'll need to better justify how counting is more fundamental than magnitude comparisons.


I'm definitely not sold on this. For one thing, "irrational" is not used as a synonym for "unknowable", it literally means "not ratio-able", i.e., cannot be expressed as a ratio. And though this is a tougher conceptual pill to swallow (ergo Pythagoras), we cannot deny that there are magnitudes that cannot be expressed as the ratio of two counting numbers.

Using our caveman ruler example from above, suppose that the finest caveman carpenter used the straightest stick and carved the most finely-spaced notches. If the finest caveman artist drew a square in the sand using the stick and carefully measured the diagonal, they would find that -- no matter how fine the notches were spaced -- the length of the diagonal would never touch a notch.

Yet, no one would disagree that the diagonal has a length. So, why should the magnitude of this length not be a number?


This needs a lot of work. Pi is not the ratio of two numbers (which, to you, are only integers). In any case, Pi (and any other transcendental number) can also be a magnitude -- indeed, 180° is precisely equal to Pi radians.


This is patently false. Let A be a set of three dogs and B a set of five cats. Then |A| < |B| is a perfectly valid (and true) statement. The cardinality of a set is the count of its members, which is a unit-less number. Therefore, the cardinalities of any sets can always be compared.

Great analysis. If you are onboard here with the following Batman and Robyn approach (or Batwoman and Robin), what say if you see 63.8% legitimacy in the writing, and you also *know* what I'm getting at, how about you proffer a re-write of the section to be more in order with academonics it is I'm trying to say? I don't want this to be just me ulilaterally pumping it out and then you're just editor... it can be co-written... ;--) I'll take those comments and incorporate a re-write shortly.

I get your point regarding numbers that are defined by arithmetic operations, and it has merit. It seems clear that the counting numbers (ℕ) are distinguished in that they're not born from arithmetic, just counting, so I'm totally with you on recognizing this important property.

But where I get iffy is in your claim that "arithmetic numbers" are not numbers.

Wow! Ok... For the record, I think they need a different term of some kind. The entire confusion here is in the co-opting of the term NUMBER cross-board.

The issue with R is that it has no actual cardinality outside of infinite, as does N. It's like saying comparing the set of discrete elements with the "set of a mathematical sine-wave." How do you rationally compare discrete to continuous in this manner? How do you compare rational to irrational? Cardinality is a RATIONAL thing. R is irrational to begin with!

 

Yes, yes, yes! The concept of SELF seems to be the defining characteristic of consciousness. Like all other concepts, the concept of SELF is a bunch of state associations. It's not located in one place.


No idea what "_REAL" means.

The difference is that H2O must have H and O, whereas computation need not be binary. Difference between necessary and sufficient.


The last sentence is fine, the rest are unjustified. We can write base-2 numbers in base-1 (or base-42). Therefore, base-2 is neither special nor privileged.


I'm fine with this, but it's what's technically known as a "weak statement": it gives a sufficient condition for computation. A strong statement demands necessary conditions, e.g., "There are computations that can only be done with a binary system." The latter statement, though strong, is provably false.

My point is in saying computation is BUILT on binary as the lowest common denominator of assumption (GRUNT). Base-48104817 is an abstraction that is BUILT from it. The proof that it *is* built from it is in the fact that a computer (or physical substrate) is using 0 and 1 (or just 1) to do ANY kind of computation, and can yield results in any base.

I'm relating the fact that in a circuit, would you agree that the voltages are present or not as the basis? Therefore, the voltage is a 1, the absence (or low, contrastive state) is a 0. OR if you built a unary system, it's just present. But in any case, there is not a 3rd or higher state of the electricity to represent anything more than GRUNT or GRUNT'. Yes?

This proof is very, very foundational in my estimation. It goes back to relating numbers to bits. We need to draw a clear connection to them, because we use bits to create numbers, and the numbers we create are based on 0's and 1's when it comes to a unary or binary system, which is the only thing the hardware is working with.

In the case of a quantum computer, we anneal the qubits into a polar state of 0 or 1, and then let them do an oscillatory dance in the Hilbert continuum dance hall, and then once measured they're yielding one of 2 knowable, "rational states" from some kind of irrational continuum belly dance.

 

Re-write:

As a function of the physical limits of any state-processing machine, the number 1 ("one") and 0 (“zero”), are the foundational numbers from which all other numbers are conceptualized due to electricity's contrastive state of high or low as the representative element of computation. Electricity is either present over a certain voltage level on a line ("one") or present below the same line ("zero"). All bases and computations are therefore abstractly constructed from these 2 numbers. As a function of physical space representation, these are the foundational numbers from which all other numbers are conceptualized.

Proof: All modern computers perform any computation of any kind using the number 1 (unary) or 1 and 0 (binary) as represented by high and low voltages as classical bits, and yield results in any base. QED.

 

LOL! You are the real MC.


That's exactly my point, it's absurd to consider something "more real" than another. When you can give me an example of something that is not "real", then I'll concede to using "real" as a meaningful word.

Perhaps "real" is like an irrational "continuum" (like ℝ! Ha!). In that we make a distinction between the "concepts" in the mind vs "real life." Clearly we all do this. Real life is "more real" than the dream state experientially. If you fall in your dreams and break your conceptual arm, that might have felt "REAL" to you, but it was NOT as "REAL" as falling off a REAL balcony and breaking your REAL arm. No?

 

Yes, yes, yes! The concept of SELF seems to be the defining characteristic of consciousness. Like all other concepts, the concept of SELF is a bunch of state associations. It's not located in one place.

But we haven't defined it. What if it's as weird as the "continuum" — and there might be a "continuum" element to the being that has nothing to do with n associated states (Cue: "soul"). ℝ as a set, for example, is really a continuum. Any interval in the continuum is 0% of the rest of it, but yet we can point to an interval, as though we can "rationalize" or "discretize" a portion as "one componental thing," but it's not a fraction of the rest of it.

I believe "REAL" can actually apply in the end to geometry that doesn't exist in "external" physical space, but the "meta-physical" space within us. That's like 6 years down the road though. LOL.

 

The cardinalities of ℕ and ℝ are a comparative illusion rendered due to "number base" zoom level. When the bases are equalized to the most elementary of base-1 “unary,” it becomes clear:

The set of ℕ represented in base-1 unary is rendered as a set of infinite marks. These are considered "countable" because they are discrete.

The set of ℝ as represented in base-1 unary is also rendered as a set of infinite marks. The difference is, these marks are grouped to create the illusion of "more marks." How many marks are in ℝ? Infinite. How many marks are in ℕ? Infinite. Why are these marks any more or less "countable?"

Both cardinalities are infinite. The difference is in the scale being used to compare any one section to another. This is only elucidated at the unary comparison level.

This is no different than the difference between discrete vs. continuous mathematics. They are not directly comparable.

The cardinality is NOT less than the other, just one set is less “grouped” and more addressable and knowable by a human. QED in my mind.

 

Great analysis. If you are onboard here with the following Batman and Robyn approach (or Batwoman and Robin), what say if you see 63.8% legitimacy in the writing, and you also *know* what I'm getting at, how about you proffer a re-write of the section to be more in order with academonics it is I'm trying to say? I don't want this to be just me ulilaterally pumping it out and then you're just editor... it can be co-written... ;--) I'll take those comments and incorporate a re-write shortly.

Do the re-write and we'll work from there.

How do you rationally compare discrete to continuous in this manner? How do you compare rational to irrational? Cardinality is a RATIONAL thing. R is irrational to begin with!

First, note that the rational numbers are dense in ℚ. When I think of discrete, I think of indivisible chunks. But between any two rational numbers p and q, we can always find another rational number, e.g., (p + q)/2. That doesn't seem discrete to me.

Second, it is entirely incorrect to say that ℝ is irrational. In fact, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. In other words, there are a whole bunch of rational (and integer and natural) numbers in ℝ.

 

My point is in saying computation is BUILT on binary as the lowest common denominator of assumption (GRUNT). Base-48104817 is an abstraction that is BUILT from it. The proof that it *is* built from it is in the fact that a computer (or physical substrate) is using 0 and 1 (or just 1) to do ANY kind of computation, and can yield results in any base.

That's not a valid proof, though. Allow me to restate your argument: Computation is fundamentally binary; we know this because computers use binary to do computations. Notice that the second part (after the semicolon) does not imply the first.

Here's an equivalently invalid argument. (In the set up, I build a three-level computer.) Computation is fundamentally ternary; we know this because this computer uses ternary computations. See how it's invalid?

I'm relating the fact that in a circuit, would you agree that the voltages are present or not as the basis?

No, I wouldn't agree. Typical computers use two states: HIGH and LOW. At the circuit level, HIGH is specified as a range of voltages (say, 0.9 to 1.5 V) and LOW is specified as another range of voltages (say, 0.0 to 0.6 V). Note that it is not necessary that LOW < HIGH, just that they are distinct. Any other voltage level results in undefined circuit behavior.

It's important to realize that the notion of "no voltage present" isn't physically meaningful. We're surrounded by the electric field, so there's always some voltage relative to a reference. This (and noise) is why we need to define ranges of acceptable voltage levels. In essence, a binary digital circuit is a map from voltage levels in ℝ to voltage levels in {LOW, HIGH}.

But in any case, there is not a 3rd or higher state of the electricity to represent anything more than GRUNT or GRUNT'. Yes?

No, we can just as easily map voltages from ℝ to voltages in {LOW, MEDIUM, HIGH} or any other number of levels.

 

Do the re-write and we'll work from there.


First, note that the rational numbers are dense in ℚ. When I think of discrete, I think of indivisible chunks. But between any two rational numbers p and q, we can always find another rational number, e.g., (p + q)/2. That doesn't seem discrete to me.

Second, it is entirely incorrect to say that ℝ is irrational. In fact, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. In other words, there are a whole bunch of rational (and integer and natural) numbers in ℝ.

Meh, they become integrated in the continuum of infinite unary marks, tho. They’re all continua, just different segmentations. If you take an interval out of infinite NZQ, you have taken out 0% of them. Same with R.

 

That's not a valid proof, though. Allow me to restate your argument: Computation is fundamentally binary; we know this because computers use binary to do computations. Notice that the second part (after the semicolon) does not imply the first.

Here's an equivalently invalid argument. (In the set up, I build a three-level computer.) Computation is fundamentally ternary; we know this because this computer uses ternary computations. See how it's invalid?


No, I wouldn't agree. Typical computers use two states: HIGH and LOW. At the circuit level, HIGH is specified as a range of voltages (say, 0.9 to 1.5 V) and LOW is specified as another range of voltages (say, 0.0 to 0.6 V). Note that it is not necessary that LOW < HIGH, just that they are distinct. Any other voltage level results in undefined circuit behavior.

It's important to realize that the notion of "no voltage present" isn't physically meaningful. We're surrounded by the electric field, so there's always some voltage relative to a reference. This (and noise) is why we need to define ranges of acceptable voltage levels. In essence, a binary digital circuit is a map from voltage levels in ℝ to voltage levels in {LOW, HIGH}.


No, we can just as easily map voltages from ℝ to voltages in {LOW, MEDIUM, HIGH} or any other number of levels.

Ok, from here on out, “presence or absence” means “high or low”.

Do computers only use logic 0 and logic 1, or is there a logic 2, a logic 5? There is only true and false represented by, in the end, high and low, right? The classical Shannon bit is that base line represented in hardware, no??

 

Here's my rewrite from scratch... I didn't bother creating axioms/theorems, etc. You have a much better sense than I do of what is most academically sound in that respect. It's just raw thoughts in a paper form. I'm sure it could be deftly propositioned if you're on board with it. ;--)

The One True Number Set

There's a reason why most mathematicians ignore the number sets. It's the same reason computers do. It's because they don't really exist as legitimate mathematical elements.

On the most elementary of levels, any modern computer does any imaginable computation and logic evaluation using the classical bit and no number sets. Because this is true, there is an overlap in utility between the classical bit, having logic state 1 and 0 and the numbers 1 and 0 as the basis of all computation.

In a computer, 1 and 0 stand for logic states HIGH or TRUE, and LOW or FALSE. A certain amount of voltage present on a line stands for logic 1, and a certain contrastive amount stands for logic 0. The transistors in IC's let voltage flow in a given direction or not. It is a binary phenomenon.

In any form of physical substrate, given sufficient complexity of arrangement of power, lines, switches and 1 or more timing devices, any form of logical evaluation and numeric computation is empirically proven to be achieved with just these two elementary states, which double as elementary "building-block" numbers.

Our ability to prove this phenomenon speaks to another related truth concerning the nature of numbers used in mathematics.

There has long been a debate concerning the cardinality of different number sets being larger or smaller. But this debate is easily settled and all confusion removed when we reduce all number sets, namely N,W,Z,Q,I,R,C to their basest of forms, and that is the "unary" base, involving just the number 1. All numbers and computation can be done with the number 1. It is not efficient, but it is elementary, and necessary to disabuse confusion.

How many values are in each of the 7 number sets when every term is represented by strings of the number 1? Infinite. What is their cardinality? Infinite.

From this perspective, it is easy to see how preposterous it is in calling the set N < R, once the positional illusion of base-10 is removed. It is due to the use of higher bases that the illusion of comparability where there is none persists. Any such manufactured sets are 100% equal in cardinality, because they are subsets of NUMBERS: no computer would argue, and most working mathematicians tacitly agree in their everyday dismissal of the 7 sets.

Once leveled by unary representation, all sets therefore implicitly disappear entirely, and we are left with one set, called NUMBERS in the unary base. This set has a most curious property: it is both countable AND uncountable at the same time. The set has no upper or lower bound, and the "density" of numbers within the set is a function of user-defined "scaling" or "zoom level." Any given interval can have as many numbers as one wants.

The same concept can be understood in the analysis of a Euclidean line with end points A and B. How many points are in between each end point? Infinite. But one can pick an interval p and q in between A and B and create another line from it. How many points are in between new line p and q? Infinite.

So how does one get a "countable number range" out of the continuous NUMBERS set?

Between any interval p and q, there are actually an infinite number of 1's or a fixed number of ones, depending on how "packed" or "unpacked" one wants to see that interval. So the concept of finitude is a function of discretizing the continuous to create computable, or "rational" value.

It could be said that the NUMBERS set is also an infinite number of classical bits set high, as a computer would represent it. A quantum computer is fed a set number of values of 1's or 0's, and within each qubit’s infinite-dimension Hilbert space using NUMBERS, mysterious computations are done of a continuous nature, and the system is constrained to yield a single NUMBER or bit per qubit. This single value is "just another bit" or NUMBER that is representing the "continuous bits" within the system.

Discretization is tied to the concept of measurability or "rationalizing" the continuous. We do this by comparing one set of infinities against another and then calling it "smaller or bigger" when the magnitude is actually identical mathematically.

A euclidean line from point A to point B has infinite points. It's only until we discretize A as a countable number and B as a countable number, and then count n numbers in between do we arrive at the concept of measurability of the continuous.

If we say a line on the Cartesian plane with any y values has an x as 3 and 10, we say there are 7 points in between. But without the numbers of 3 and 10, how many actual points are in between using the NUMBERS set? Infinite.

It's therefore necessary to understand "measurability" or "magnitude" as a strange comparative illusion. A number has no size until it is compared with another, but even then, this could be argued as arbitrary, because the concept of "after or before" in a sequence speaks to ordinality only, and the TOTAL number of terms speaks to cardinality. In this case, INFINITE cardinality doesn't help. So something "more" must be attributed to imbue the sense of magnitude in physical space.

One can label a stick of length 6 vs. length 3, and in unary that is IIIIII vs. III. Which stick is bigger than the other? Sure, 6 comes "after" 3's ordinality in the NUMBERS set, but with a set cardinality of INFINITY, does that mean the stick is "bigger?" No.

It's only until we assign an appendix to the number in physical space does the sense of magnitude exist. We do this by creating a physical-space RULER of some kind, where the ruler is segmented proportionally into units—for example, inches.

But how big is an inch? We do not know until we compare it to something else. 1 inch is "how long?" It's the same problem as trying to measure the stick! 1 inch could be divided into 2.5 cm, but how long is a cm? 10 mm. In the mind, one could continue to divide this and create new names for what would be consider "smaller measurements (even though smaller is not defined!). In physical space, we are limited by physics.

So a stick of 6 inches is "bigger in magnitude" than a stick of 3 inches only because we've defined an inch in terms of something else. But why is attributing an inch different than a cm or any other thing that we've measured in terms of a number we've assigned to it? This is a grand illusion!

Magnitude is a recursive infinite "turtles all the way down" phenomena that terminates somewhere within our being in order to create the "sense" of comparative "magnitude." It is here that we might invoke a term called "FEELING" to arrive at a SENSE of the MAGNITUDE. Feeling is perhaps an innate experiential-based measurement capacity which allows us to discretize the sense of endless numeric comparison continuum into something that feels bigger or smaller innately—not because of numbers alone, but because one feels different with one vs. another, and this comparative element provides the illusion of magnitude from numbers that actually don't have any in and of themselves.

 

Perhaps, just perhaps! Speaking of definitions for REAL, Des Cartes wasn’t wrong after all! There is only one “set“ of numbers... the REAL one, which is a synonym for “all numbers!” And the rest are “_MYTH” or “_FAKE!” Lol!

 

Perhaps "real" is like an irrational "continuum" (like ℝ! Ha!). In that we make a distinction between the "concepts" in the mind vs "real life." Clearly we all do this. Real life is "more real" than the dream state experientially. If you fall in your dreams and break your conceptual arm, that might have felt "REAL" to you, but it was NOT as "REAL" as falling off a REAL balcony and breaking your REAL arm. No?

Notice how vague all of this is for a presumably straightforward notion. Also notice that it presumes a distinction between "mind" and "reality", neither of which have any meaning in my model!

Try to distill what you're trying to say here into precise expressions. To me, it seems like you want "real" to express the degree to which something is empirically corroborated, but that is a slippery slope into perceptual solipsism.

Maybe the way to frame the issue is: What formal purpose does token "real" serve, i.e., what phenomena of experience does "real" explain? However you answer this question, see if there is a simpler, clearer way to explain the phenomenon.

 

Notice how vague all of this is for a presumably straightforward notion. Also notice that it presumes a distinction between "mind" and "reality", neither of which have any meaning in my model!

Try to distill what you're trying to say here into precise expressions. To me, it seems like you want "real" to express the degree to which something is empirically corroborated, but that is a slippery slope into perceptual solipsism.

Maybe the way to frame the issue is: What formal purpose does token "real" serve, i.e., what phenomena of experience does "real" explain? However you answer this question, see if there is a simpler, clearer way to explain the phenomenon.

I'm entirely spit-balling extra-model here, for the record! It all is straightforward once we figure it out, but "getting there" might be not "straight forward" at all, going down n paths to bump into m walls to find q hints about a few core simple elements.

You don't have a model for mind and reality, of course, but I personally posit you (read: every human) might implicitly have one that may be tacitly assumed on a subconscious level. For example, I believe you remarked you don't know if mathematical objects exist or not, and perhaps I can convince you (or something on that order)... perhaps that is part of the definition.

I will think on definition for "real" and see if I can put something together that may or may not involve MC Hilbert.