Just 1 reply this time. It is rather long. But some of that is due to formatting.
You seem to be randomly flipping between talking about standard infinite sets and your own concept of "literal infinite" (which I still think is a confusing name), I'll just use LI for now. In the example I was responding to, you used ℕ (strictly, you actually used {ℕ}, which is a set with one element, that element being the set of natural numbers) and I was making the point that you could add other members to the set to make a different set with the same cardinality. This example really had nothing to do with LI, I was just addressing your frequent use of ∞ + ∞ = ∞ as if this meant that you, generally speaking, couldn't add to an infinite set to get anything different, which you obviously can.
No. I'm not flipping. I'm just addressing your strawmen in the order that you are presenting them. And I'm saying 1) I'm not defining a set. 2) Even if I were, the objection is invalid.
When I say "∞ + ∞ = ∞" I'm talking about a quantity that is produced from adding numbers. It's an example.
And I'm saying "∞ + ∞ = ∞" can be generalized if each version of "∞" is identical. Consistency is required. Consistency is included in the statement.
I understand |ℕ| =/= |ℝ| but that fact is irrelevant to "∞ + ∞ = ∞"
I'm not saying "|ℕ|+|ℝ|=|ℕ|", that's false.
I'm not saying "|ℕ|+|ℕ|=|ℝ|", that's false.
When I say "∞ + ∞ = ∞" I am saying:
"∞ + ∞ = ∞", this is true.
|ℕ|+|ℕ|=|ℕ|, this is true.
|ℝ|+|ℝ|=|ℝ|, this is true.
Please notice the consistency above.
It get's a little wonky if |ℤ| and |ℂ| are introduced since |ℤ|=|ℕ| and |ℂ|=|ℝ|. But, it's still consistently true.
I'm saying:
"∞ + ∞ = ∞", this is true.
Countable-infinity + Countable-infinity = Countable-infinty, this is true.
Uncountable-infinity + Uncountable-infinity = Uncountable-infinity, this is true.
I'm NOT saying:
Countable-infinity + Countable-infinity = Uncountable-infinty, this is false.
Uncountable-infinity + Countable-infinity = Countable-infinty, this is false.
So, even if |ℤ| and |ℂ| are introduced, what I'm saying is still true, if the consistency is maintained. "∞ + ∞ = ∞" includes that consistency.
When you flip what I'm saying from something which is consistent into something which is not consistent, your're flipping what I'm saying, which is true, into something that is false.
What I'm saying is true.
What you're saying is false.
You're changing what I'm saying into something that's false.
Please don't change what I'm saying.
From this fact, "∞ + ∞ = ∞", I can logically state: "a + ∞ = ∞", where 'a' is anything included in '∞'.
Included.
So... when you start flipping what I'm saying into something inconsistent, ( where a is not included ) I flip it back into something that is true.
If you want to consider |ℕ| and |ℝ|, because |ℕ| =/= |ℝ|, then I say:
"a + ∞ = ∞", which is true
|ℕ| + |ℝ| = |ℝ|, which is true
If you claim I'm saying "|ℕ|+|ℝ|=|ℕ|", I will repeatedly tell you, "I'm NOT saying that" |ℝ| is NOT included in |ℕ|. Don't change what I'm saying.
(And, anytime you want to consider an aleph number or a beth/bet number or any cardinality of an infinite set.. that is { |the-infinite-set| }. That's it. A singleton. Nothing more, nothing less.)
What's the significant difference? They are concepts in the category of 'relationships'.
In category theory, there's objects, morphisms, and functors. Morphisms and functors are intra-category relationships and inter-category relationships. In the super-category of all categories, there will be objects describing the intra/inter-category relationships. That's what you're calling a concept. But this does not replace or duplicate the morphisms/functors that are intrinsic to the super-category's structure.
And eventhough I'm using the category theory framework, I'm really just describing a relational database and object oriented programming languages. It's a framework. A structure. Those relationships are important to the defiintion.
It's important when discussing nullification, because the nullifcation is a/∞. 'a' is not nullified, but the relationship is. An absolutely omnipotent, literally infinite, being creating something distinct and other than itself is a relationship. That relationship is not possible, yet.
Again, you seem to be flipping from concept to concept without keeping track. You have made a great deal of this ∞ + ∞ = ∞ throughout as an example of why you can't add to infinity to get anything else, but generally speaking, you obviously can. You may argue that you can't with LI but that is entirely non-standard and just your own idea. You need to be clear what it is you're talking about.
I am not flipping. I am following your strawmen. You can't add anything which is
included in infinity to infinity. It's already included. When you try to add something which is not included, that's an irrelevant strawman.
Now we're suddenly back to nullification - they are infinitesimal which is not the same a non-existent.
It's not sudden. Nullification is the point. It's what you are objecting to.
They are infinitesimal when considered as included. They are non-existent when considered as excluded.
From the perspective of an included object, they, the relationships, are infinitesimal as all-inclusion is approached.
From the perspective of an excluded object, eventhough this violates the defintion, they, the relationships, are completely non-existent, because Literal-Infinity
IS all-inclusive.
This proves that the definition is logically consistent.
"a/∞ =0" AND "lim(x-->∞) a/x-->0"
If 'a' is a new creation, "a/∞ =0"
If 'a' is included in 'x', "lim(x-->∞) a/x-->0"
So, there it is ^^. A logical statement and definition. If you are saying it is illogical, it is your burden to show that.
First, a procedure isn't a definition.
Sure a procedure is a definition. Define a sine-wave. "A sine wave is what happens when..."
Second, and as I explained before, since your procedure involves systematically adding lists of things, I really don't see how it's going to produce anything other than a standard set with cardinality ℵ₀. What you need is a proper, rigorous definition.
There are several steps in the definition which establishes relationships between all the objects. Certainly you can see the difference between counting and developing a database.
You're the one who brought up the 2-d plane because you didn't like my 1-d function. If it was irrelevant, why mention it?
Let's see... let me scroll back and figure that out. ... .... .... OK!
The point I was making several posts back was that the cardinality is a one dimensional attribute. The set which produces it only has 1 relationship. 1 function. And that function produces 1 thing, it's not even a collection. It's barely a set. It's very very different from what I'm describing. But, if you want to consider it, the same nullification occurs as long as consistency is maintained.
And for those of us looking for logic expressed in English or mathematics....?
I've given it you. You keep changing it.
In english:
LI:
absolutely all-inclusive relational database
Nullification:
"external similarities and differences between objects and LI" = 0
"internal similarities/differences between objects and LI" = infinitesimal
In logic:
LI ----> Nullification
See:
en.wikipedia.org
You are proposing something (nullification) with is entirely new (as far as I can see) and is contradicted by literally everything else that uses either limit infinity or actual transfinite numbers.. It really isn't up to me to disprove it, the burden of proof is entirely yours. So far, all you have is some hand-waving, using a fairly standard limit process, like here, and some vague and largely irrelevant stuff about ∞ + ∞ = ∞.
No.... you haven't brought anything that contradicts the definitions. And the burden is on you to do so. You're saying that I have brought something which is illogical. You have the burden to show that. We had an agreement; you would consider the god concept I defined ,if it was not contradictory.
Please make a list of the contradictions you brought. You don't need to explain them, just a simple list. And then I'll breifly list each of my rebuttals. Throughout the conversation, you haven't undermined any of my rebuttals. Sometimes you'll say "nonsense" or "word salad". Then I'll explain the revelance and the "sense" of what I'm saying. And nothing more comes of it. That's why it's time to move on unless you can bring anything new which is relelvant and valid.
Please make a list. And then I'll very briefly attach my rebuttals. Then you can rebut the rebuttals. But, be ready to see "this is changing the definition" repeatedly.
It really isn't. That's just making the elements infinitesimal compared to the whole. This is entirely standard and you're trying to force an entirely non-standard interpretation onto it.
No. That would be the quantity of the included objects compared to the quanitity of all the included objects is infinitesimal. The included objects themselves remain unchanged.
lim(x-->∞) a/x-->0 is describing 'a/x'. It is not describing 'a'
a/∞ is describing 'a/∞' not 'a'
'a' is not changing. Nullification is a ratio, *points to your screen name*
What this means in english, how it is interpretted depends on how 'a' and 'x' and '∞' are defined.
I have defined 'x' as a relational database.
I have defined '∞' as all-inclusion
I have defined 'a' as the similarities and differences between the included objects in the relational database.
This results in absolute nullification of any similarities and differences which are excluded. This is consistent with the definition.
This results in approaching absolute nullification of any similarities and differences which are included. This is consistent with the definition.