What is the difference between actual infinity and potential infinity?

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• 401
The Wittgensteinian intuition regarding the identity sign, roughly put, and defended here by MU, is that it is the meta-logical expression of synonymy, which upon full analysis of the expression concerned, is eliminated to yield substitution operations among 'non equal' logical terms, each denoting distinguishable objects. This point of view has been shown to be as expressive and as consistent as the logical interpretation of identity, even if on occasion standard theorems of mathematics need amending.
• 32
There is no "k" though. What is symbolized is "2+2", two objects and an operator, not one object "k". So this object represented by "k" is not represented by "2+2", it has been wrongly created by you mind, false imagination, nothing here represents it.

I introduced the object k as an intermediate clarification step, of course it is not mentioned by 2+2.

I just want you to answer this question

does the expression "2 + 2" denotes two objects or one object?

I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects"

Take the following example: "The planet between planets Venus and Mars"

Obviously this expression contains expressions "Venus", "Mars" and each is denoting an object. So it does contain denotations of two objects in it. BUT it itself denotes ONE object that is the object denoted by expression Earth. What we mean by "denote" here is the object that is the subject of speech of that expression, which is obviously the object denoted by expression Earth in English. This is an example of an expression that contains denotation of more than one object within it, but it itself only denotes ONE object.

In a similar manner 2 + 2 is denoting ONE object.

2 + 2 is equivalent to the expression "The result of summation of 2 and 2"

Or sometimes we express it as "The sum of 2 and 2"

2 + 2 means "the natural number that results from adding 2 to 2"

So '2 + 2' is denoting a single object, although it does contain inside it two occurrences of a denotation, yet it is denoting a single object, similarly 2+3 it contains two distinct denotations, but it itself is denoting one object which is the object that results from adding 2 to 3.

Now whether the SINGLE object denoted by '2 + 2' is itself the same (identical to the) object that is denoted by '4', is something that I personally think it to be the easier and simpler way to formalize. If we say No, the object that 2 + 2 is denoting is different from the object 4 is denoting but it is "equal" to that object, and here equality can be understood as a kind of equivalence relation (a relation that is reflexive, symmetric and transitive), I think this is a more complicated way of looking at it.

Of course the interpretation of = as equality relation is weaker (logically speaking) than interpreting it as identity relation. Many times people prefer or feel more safe with holding weaker assumptions. And so it indeed can be justified as a kind of cautious philosophical approach to the matter. However, I still think that identity, albeit being a stronger interpretation, yet it is much nicer and sharper, and actually much easier formally speaking than the more general equality notion.

Try to formalize PA yourself using "=" as an equality relation. You'll see how cumbersome it would be. Interpreting "=" as identity simplify formalization to a great extent.
• 32
meta-logical expression of synonymy, which upon full analysis of the expression concerned, is eliminated to yield substitution operations among 'non equal' logical terms, each denoting distinguishable objects.sime

Agreed. And I mentioned this to MU. I said that one can indeed interpret the '=' sign as some equivalence relation, no doubt, like that of synonymy, or actually any equivalence relation, of course this can formally work. But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. Philosophically speaking one might prefer to hold to the weaker interpretation, but formally speaking, it is not the preferable one. For example how would you DEFINE 4. Using identity I don't need to introduce 4 as a primitive symbol, since I can define it, since it is the unique object that 2 + 2 is denoting. However you cannot define it as such when "=" is just an equivalence relation, you'll need to introduce 4 as a primitive notion, that said you'll need to introduce all naturals as primitive constants of the language, which is in some sense cumbersome.
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