Superstructure
المؤلف:
Hurd, A. E. and Loeb, P. A.
المصدر:
Ch. 3 in An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.
الجزء والصفحة:
...
13-2-2022
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Superstructure
In nonstandard analysis, the limitation to first-order analysis can be avoided by using a construction known as a superstructure. Superstructures are constructed in the following manner. Let 
be an arbitrary set whose elements are not sets, and call the elements of 
"individuals." Define inductively a sequence of sets with 
and, for each natural number 
,
and let
Then 
is called the superstructure over 
. An element of 
is an entity of 
.
Using the definition of ordered pair provided by Kuratowski, namely ![(a,b)=<span style=]()
{{a},{a,b}}" src="https://mathworld.wolfram.com/images/equations/Superstructure/Inline9.svg" style="height:22px; width:155px" />, it follows that 
for any 
. Therefore, 
, and for any function 
from 
into 
, we have 
. Now assume that the set 
is (in one-to-one correspondence with) the set of real numbers 
, and then the relation 
which describes continuity of a function at a point is a member of 
. Careful consideration shows that, in fact, all the objects studied in classical analysis over 
are entities of this superstructure. Thus, first-order formulas about 
are sufficient to study even what is normally done in classical analysis using second-order reasoning.
To do nonstandard analysis on the superstructure 
, one forms an ultrapower of the relational structure 
. Los' theorem yields the transfer principle of nonstandard analysis.
REFERENCES
Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics. New York: Academic Press, p. 16, 1986.
Hurd, A. E. and Loeb, P. A. Ch. 3 in An Introduction to Nonstandard Real Analysis. New York: Academic Press, 1985.
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