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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| Những tác giả chính: | , , |
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| Định dạng: | Sách |
| Ngôn ngữ: | English |
| Được phát hành: |
IOS Press
2013
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| Những chủ đề: | |
| Truy cập trực tuyến: | http://scholar.dlu.edu.vn/thuvienso/handle/DLU123456789/35161 |
| Các nhãn: |
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| Thư viện lưu trữ: | Thư viện Trường Đại học Đà Lạt |
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| Tóm tắt: | 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. |
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