Re: Is it sure that any mechanism has a largest model? (LI,8C, p.205)

[Judith Rosen <***>]

Hi Tim,

 

What you refer to with transfinite numbers and different sizes of infinities is exactly what happens when the semantics are dispensed with.

Regarding "THE intersection of an infinite set of models"... think about it: There is no singular "largest model" in an infinite set. There is no one "intersection". However, I can answer one aspect of that question: Any intersection of two (or more) models is also a model. In an infinite set, there will be infinite intersections.

 

Judith

----- Original Message -----

From: Tim Gwinn

To: ***

Sent: Sunday, December 12, 2004 11:20 AM

Subject: Re: [ROSEN] Is it sure that any mechanism has a largest model? (LI,8C, p.205)

Judith,

 

1) In Cantor's theory of transfinite numbers he determined that there are different....well..."sizes" or cardinalities...of infinities. Countable infinities are those such as the natural numbers or integers. Their cardinality is typically represented by the symbol aleph-zero: À₀

 

Uncountable infinites are "larger", and the real numbers are an example of that. They are of the order:  À₁

 

Wikipedia has a decent short summary:

http://en.wikipedia.org/wiki/Transfinite_number

 

 

2) I agree with your comments. But I am not sure how to prove that the intersection of an infinite set of models is itself a model. This would be a very good question for Aloisius.

 

Regards,

Tim

 

-----Original Message-----
From: ROSEN Forum [mailto:***On Behalf Of Judith Rosen
Sent: Sunday, December 12, 2004 10:20 AM
To: ***
Subject: Re: Is it sure that any mechanism has a largest model? (LI,8C, p.205)

What is a "countable infinity"?

 

Secondly: If you have a finite set of models, and each intersection of models constitutes another model, you will end up with a really huge number of models, but it would still be a finite set, would it not? Because you would exhaust the number of different combinations, eventually, before repeating them.

 

On the other hand, an infinite set of models will always be infinite.

Boris: Nevertheless, don't you think we still need to proove that the infinite intersection of models is still a model?

TG: Hmmm.....I am not sure. I think if we agree that the intersection of any finite number of models is a model, then by induction so will the intersection of an infinite number of models....at least when it is a countable infinity.

I will have to think about this some more. At the moment, I cannot think of stronger reasoning than that.