Andrzej Mostowski and Foundational Studies
The standard definition of limz→∞ F(z) = ∞ is an ∀∃∀ sentence. Mostowski showed that in the standard model of arithmetic, these quantifiers cannot be eliminated. But Abraham Robinson showed that in the nonstandard setting, this limit property for a standard function F is equivalent to the one qua...
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IOS Press
2013
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oai:scholar.dlu.edu.vn:DLU123456789-351612014-01-20T00:09:45Z Andrzej Mostowski and Foundational Studies Ehrenfeucht, A Marek, V.W Srebrny, M Foundational Study The standard definition of limz→∞ F(z) = ∞ is an ∀∃∀ sentence. Mostowski showed that in the standard model of arithmetic, these quantifiers cannot be eliminated. But Abraham Robinson showed that in the nonstandard setting, this limit property for a standard function F is equivalent to the one quantifier statement that F(z) is infinite for all infinite z. In general, the number of quantifier blocks needed to define the limit depends on the underlying structureMin which one is working. Given a structureMwith an ordering, we add a new function symbol F to the vocabulary ofMand ask for the minimum number of quantifier blocks needed to define the class of structures (M, F) in which limz→∞ F(z) = ∞ holds. We show that the limit cannot be defined with fewer than three quantifier blocks when the underlying structureMis either countable, special, or an o-minimal expansion of the real ordered field. But there are structuresMwhich are so powerful that the limit property for arbitrary functions can be defined in both two-quantifier forms. 2013-08-22T01:27:08Z 2013-08-22T01:27:08Z 2008 Book 978-1-58603-782-6 http://scholar.dlu.edu.vn/thuvienso/handle/DLU123456789/35161 en application/pdf IOS Press |
| institution |
Thư viện Trường Đại học Đà Lạt |
| collection |
Thư viện số |
| language |
English |
| topic |
Foundational Study |
| spellingShingle |
Foundational Study Ehrenfeucht, A Marek, V.W Srebrny, M Andrzej Mostowski and Foundational Studies |
| description |
The standard definition of limz→∞ F(z) = ∞ is an ∀∃∀ sentence.
Mostowski showed that in the standard model of arithmetic, these quantifiers cannot
be eliminated. But Abraham Robinson showed that in the nonstandard setting,
this limit property for a standard function F is equivalent to the one quantifier statement
that F(z) is infinite for all infinite z. In general, the number of quantifier
blocks needed to define the limit depends on the underlying structureMin which
one is working. Given a structureMwith an ordering, we add a new function symbol
F to the vocabulary ofMand ask for the minimum number of quantifier blocks
needed to define the class of structures (M, F) in which limz→∞ F(z) = ∞
holds.
We show that the limit cannot be defined with fewer than three quantifier blocks
when the underlying structureMis either countable, special, or an o-minimal expansion
of the real ordered field. But there are structuresMwhich are so powerful
that the limit property for arbitrary functions can be defined in both two-quantifier
forms. |
| format |
Book |
| author |
Ehrenfeucht, A Marek, V.W Srebrny, M |
| author_facet |
Ehrenfeucht, A Marek, V.W Srebrny, M |
| author_sort |
Ehrenfeucht, A |
| title |
Andrzej Mostowski and Foundational Studies |
| title_short |
Andrzej Mostowski and Foundational Studies |
| title_full |
Andrzej Mostowski and Foundational Studies |
| title_fullStr |
Andrzej Mostowski and Foundational Studies |
| title_full_unstemmed |
Andrzej Mostowski and Foundational Studies |
| title_sort |
andrzej mostowski and foundational studies |
| publisher |
IOS Press |
| publishDate |
2013 |
| url |
http://scholar.dlu.edu.vn/thuvienso/handle/DLU123456789/35161 |
| _version_ |
1757667271928446976 |