Intuition behind the Idealization Axiom of Internal Set Theory
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Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (non-standard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets."
Given that standard sets are supposed to represent the finite sets that we could plausibly construct, it makes sense that there would only be a finite number of them and that their union would therefore be finite. However, I can't understand why we can choose this set such that it satisfies any internal property shared by all standard finite sets. Why is this the case?
set-theory nonstandard-analysis
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
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add a comment |
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Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (non-standard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets."
Given that standard sets are supposed to represent the finite sets that we could plausibly construct, it makes sense that there would only be a finite number of them and that their union would therefore be finite. However, I can't understand why we can choose this set such that it satisfies any internal property shared by all standard finite sets. Why is this the case?
set-theory nonstandard-analysis
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
$endgroup$
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Wikipedia? Your link does not direct to Wikipedia.
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– Asaf Karagila♦
Dec 5 '18 at 12:39
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@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44
add a comment |
$begingroup$
Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (non-standard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets."
Given that standard sets are supposed to represent the finite sets that we could plausibly construct, it makes sense that there would only be a finite number of them and that their union would therefore be finite. However, I can't understand why we can choose this set such that it satisfies any internal property shared by all standard finite sets. Why is this the case?
set-theory nonstandard-analysis
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
$endgroup$
Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (non-standard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets."
Given that standard sets are supposed to represent the finite sets that we could plausibly construct, it makes sense that there would only be a finite number of them and that their union would therefore be finite. However, I can't understand why we can choose this set such that it satisfies any internal property shared by all standard finite sets. Why is this the case?
set-theory nonstandard-analysis
set-theory nonstandard-analysis
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
edited Dec 5 '18 at 12:43
Casebash
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
asked Dec 5 '18 at 11:20
CasebashCasebash
5,66844271
5,66844271
$begingroup$
Wikipedia? Your link does not direct to Wikipedia.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 12:39
$begingroup$
@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44
add a comment |
$begingroup$
Wikipedia? Your link does not direct to Wikipedia.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 12:39
$begingroup$
@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44
$begingroup$
Wikipedia? Your link does not direct to Wikipedia.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 12:39
$begingroup$
Wikipedia? Your link does not direct to Wikipedia.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 12:39
$begingroup$
@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44
$begingroup$
@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44
add a comment |
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$begingroup$
Wikipedia? Your link does not direct to Wikipedia.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 12:39
$begingroup$
@AsafKaragila: Oops, that's the result of a Chrome plugin that reformats Wikipedia
$endgroup$
– Casebash
Dec 5 '18 at 12:44