Double jumps of minimal degrees
This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree a < 0′ satisfies a ″ = 0″ . To restate this result in more suggestive language and compare it with related results, we shall use notation based...
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| Veröffentlicht in: | The Journal of symbolic logic 1978-12, Vol.43 (4), p.715-724 |
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| container_title | The Journal of symbolic logic |
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| creator | Jockusch, Carl G. Posner, David B. |
| description | This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree
a
< 0′
satisfies
a
″
=
0″
. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let
H
n
be the class of degrees
a
< 0′
such that
a
(n)
=
0
(n+1)
, and let
L
n
be the class of degrees
a
≤
0′
such that
a
n
=
0
n
. (Observe that
H
i
⊆
L
j
and
L
i
⊆
L
j
, whenever
i
≤
j
, and
H
i
∩
L
j
= ∅ for all
i
and
j.
) The result mentioned above may now be restated in the form that every minimal degree
a
≤ 0′ is in
L
2
. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree
a
< 0′ is in
H
1
. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree
a
< 0′
which is not in
L
1
. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree
a
< 0′
in
L
1
. Thus each minimal degree
a < 0′
lies in exactly one of the two classes
L
1
L
2
–
L
1
and each of the classes contains minimal degrees.
Our results are not restricted to the degrees below
0′
. We show in fact that every minimal degree
a
satisfies
a″
= (
a
∪
0′
)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below
0′
to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let
GH
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n)
, and let
GL
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n−1)
. |
| doi_str_mv | 10.2307/2273510 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_proje</sourceid><recordid>TN_cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183740317</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>2273510</jstor_id><sourcerecordid>2273510</sourcerecordid><originalsourceid>FETCH-LOGICAL-c373t-3c5b7cc3a4097e931a67dc507df64c3f7649be2149f88c67c546105bbdc2f6873</originalsourceid><addsrcrecordid>eNp9kEtLxDAUhYMoOI7iT7Cg4Kp68052StVRKfjAWYc2TaW1Mx2TFvTf26Fllq4OXD7OOfcgdIrhilCQ14RIyjHsoRnWjMZcKbGPZgCExExhcoiOQqgBgGumZujsru3zxkV1v9qEqC2jVbWuVlkTFe7TOxeO0UGZNcGdTDpHy4f7j-QxTl8WT8ltGlsqaRdTy3NpLc0YaOk0xZmQheUgi1IwS0spmM4dwUyXSlkhLWcCA8_zwpJSKEnn6Gb03fi2drZzvW2qwmz8UMb_mjarTLJMp-skdWgMxopKBhRvLc53Ft-9C52p296vh9YGEz38C1zAQF2OlPVtCN6VuwwMZjugmQYcyIuRrEPX-n-weMSq0LmfHZb5LyMkldyIxZvRgr-mRD6bd_oHlTZ6Gw</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1290590560</pqid></control><display><type>article</type><title>Double jumps of minimal degrees</title><source>Periodicals Index Online</source><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><creator>Jockusch, Carl G. ; Posner, David B.</creator><creatorcontrib>Jockusch, Carl G. ; Posner, David B.</creatorcontrib><description><![CDATA[This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree
a
< 0′
satisfies
a
″
=
0″
. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let
H
n
be the class of degrees
a
< 0′
such that
a
(n)
=
0
(n+1)
, and let
L
n
be the class of degrees
a
≤
0′
such that
a
n
=
0
n
. (Observe that
H
i
⊆
L
j
and
L
i
⊆
L
j
, whenever
i
≤
j
, and
H
i
∩
L
j
= ∅ for all
i
and
j.
) The result mentioned above may now be restated in the form that every minimal degree
a
≤ 0′ is in
L
2
. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree
a
< 0′ is in
H
1
. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree
a
< 0′
which is not in
L
1
. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree
a
< 0′
in
L
1
. Thus each minimal degree
a < 0′
lies in exactly one of the two classes
L
1
L
2
–
L
1
and each of the classes contains minimal degrees.
Our results are not restricted to the degrees below
0′
. We show in fact that every minimal degree
a
satisfies
a″
= (
a
∪
0′
)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below
0′
to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let
GH
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n)
, and let
GL
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n−1)
.]]></description><identifier>ISSN: 0022-4812</identifier><identifier>EISSN: 1943-5886</identifier><identifier>DOI: 10.2307/2273510</identifier><language>eng</language><publisher>New York, USA: Cambridge University Press</publisher><subject>Degree of unsolvability ; Eigenfunctions ; Mathematical functions ; Mathematical logic ; Mathematical sets ; Recursive functions</subject><ispartof>The Journal of symbolic logic, 1978-12, Vol.43 (4), p.715-724</ispartof><rights>Copyright 1979 Association for Symbolic Logic, Inc.</rights><rights>Copyright 1978 Association for Symbolic Logic</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c373t-3c5b7cc3a4097e931a67dc507df64c3f7649be2149f88c67c546105bbdc2f6873</citedby><cites>FETCH-LOGICAL-c373t-3c5b7cc3a4097e931a67dc507df64c3f7649be2149f88c67c546105bbdc2f6873</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2273510$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2273510$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,27869,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>Jockusch, Carl G.</creatorcontrib><creatorcontrib>Posner, David B.</creatorcontrib><title>Double jumps of minimal degrees</title><title>The Journal of symbolic logic</title><description><![CDATA[This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree
a
< 0′
satisfies
a
″
=
0″
. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let
H
n
be the class of degrees
a
< 0′
such that
a
(n)
=
0
(n+1)
, and let
L
n
be the class of degrees
a
≤
0′
such that
a
n
=
0
n
. (Observe that
H
i
⊆
L
j
and
L
i
⊆
L
j
, whenever
i
≤
j
, and
H
i
∩
L
j
= ∅ for all
i
and
j.
) The result mentioned above may now be restated in the form that every minimal degree
a
≤ 0′ is in
L
2
. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree
a
< 0′ is in
H
1
. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree
a
< 0′
which is not in
L
1
. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree
a
< 0′
in
L
1
. Thus each minimal degree
a < 0′
lies in exactly one of the two classes
L
1
L
2
–
L
1
and each of the classes contains minimal degrees.
Our results are not restricted to the degrees below
0′
. We show in fact that every minimal degree
a
satisfies
a″
= (
a
∪
0′
)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below
0′
to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let
GH
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n)
, and let
GL
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n−1)
.]]></description><subject>Degree of unsolvability</subject><subject>Eigenfunctions</subject><subject>Mathematical functions</subject><subject>Mathematical logic</subject><subject>Mathematical sets</subject><subject>Recursive functions</subject><issn>0022-4812</issn><issn>1943-5886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1978</creationdate><recordtype>article</recordtype><sourceid>K30</sourceid><recordid>eNp9kEtLxDAUhYMoOI7iT7Cg4Kp68052StVRKfjAWYc2TaW1Mx2TFvTf26Fllq4OXD7OOfcgdIrhilCQ14RIyjHsoRnWjMZcKbGPZgCExExhcoiOQqgBgGumZujsru3zxkV1v9qEqC2jVbWuVlkTFe7TOxeO0UGZNcGdTDpHy4f7j-QxTl8WT8ltGlsqaRdTy3NpLc0YaOk0xZmQheUgi1IwS0spmM4dwUyXSlkhLWcCA8_zwpJSKEnn6Gb03fi2drZzvW2qwmz8UMb_mjarTLJMp-skdWgMxopKBhRvLc53Ft-9C52p296vh9YGEz38C1zAQF2OlPVtCN6VuwwMZjugmQYcyIuRrEPX-n-weMSq0LmfHZb5LyMkldyIxZvRgr-mRD6bd_oHlTZ6Gw</recordid><startdate>19781201</startdate><enddate>19781201</enddate><creator>Jockusch, Carl G.</creator><creator>Posner, David B.</creator><general>Cambridge University Press</general><general>Association for Symbolic Logic, Inc</general><general>Association for Symbolic Logic</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>EOLOZ</scope><scope>FKUCP</scope><scope>IOIBA</scope><scope>K30</scope><scope>PAAUG</scope><scope>PAWHS</scope><scope>PAWZZ</scope><scope>PAXOH</scope><scope>PBHAV</scope><scope>PBQSW</scope><scope>PBYQZ</scope><scope>PCIWU</scope><scope>PCMID</scope><scope>PCZJX</scope><scope>PDGRG</scope><scope>PDWWI</scope><scope>PETMR</scope><scope>PFVGT</scope><scope>PGXDX</scope><scope>PIHIL</scope><scope>PISVA</scope><scope>PJCTQ</scope><scope>PJTMS</scope><scope>PLCHJ</scope><scope>PMHAD</scope><scope>PNQDJ</scope><scope>POUND</scope><scope>PPLAD</scope><scope>PQAPC</scope><scope>PQCAN</scope><scope>PQCMW</scope><scope>PQEME</scope><scope>PQHKH</scope><scope>PQMID</scope><scope>PQNCT</scope><scope>PQNET</scope><scope>PQSCT</scope><scope>PQSET</scope><scope>PSVJG</scope><scope>PVMQY</scope><scope>PZGFC</scope></search><sort><creationdate>19781201</creationdate><title>Double jumps of minimal degrees</title><author>Jockusch, Carl G. ; Posner, David B.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c373t-3c5b7cc3a4097e931a67dc507df64c3f7649be2149f88c67c546105bbdc2f6873</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1978</creationdate><topic>Degree of unsolvability</topic><topic>Eigenfunctions</topic><topic>Mathematical functions</topic><topic>Mathematical logic</topic><topic>Mathematical sets</topic><topic>Recursive functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jockusch, Carl G.</creatorcontrib><creatorcontrib>Posner, David B.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Periodicals Index Online Segment 01</collection><collection>Periodicals Index Online Segment 04</collection><collection>Periodicals Index Online Segment 29</collection><collection>Periodicals Index Online</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - West</collection><collection>Primary Sources Access (Plan D) - International</collection><collection>Primary Sources Access & Build (Plan A) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Midwest</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Northeast</collection><collection>Primary Sources Access (Plan D) - Southeast</collection><collection>Primary Sources Access (Plan D) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Southeast</collection><collection>Primary Sources Access (Plan D) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - UK / I</collection><collection>Primary Sources Access (Plan D) - Canada</collection><collection>Primary Sources Access (Plan D) - EMEALA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - International</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - International</collection><collection>Primary Sources Access (Plan D) - West</collection><collection>Periodicals Index Online Segments 1-50</collection><collection>Primary Sources Access (Plan D) - APAC</collection><collection>Primary Sources Access (Plan D) - Midwest</collection><collection>Primary Sources Access (Plan D) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Canada</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - EMEALA</collection><collection>Primary Sources Access & Build (Plan A) - APAC</collection><collection>Primary Sources Access & Build (Plan A) - Canada</collection><collection>Primary Sources Access & Build (Plan A) - West</collection><collection>Primary Sources Access & Build (Plan A) - EMEALA</collection><collection>Primary Sources Access (Plan D) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - Midwest</collection><collection>Primary Sources Access & Build (Plan A) - North Central</collection><collection>Primary Sources Access & Build (Plan A) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - Southeast</collection><collection>Primary Sources Access (Plan D) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - APAC</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - MEA</collection><jtitle>The Journal of symbolic logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jockusch, Carl G.</au><au>Posner, David B.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Double jumps of minimal degrees</atitle><jtitle>The Journal of symbolic logic</jtitle><date>1978-12-01</date><risdate>1978</risdate><volume>43</volume><issue>4</issue><spage>715</spage><epage>724</epage><pages>715-724</pages><issn>0022-4812</issn><eissn>1943-5886</eissn><abstract><![CDATA[This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree
a
< 0′
satisfies
a
″
=
0″
. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let
H
n
be the class of degrees
a
< 0′
such that
a
(n)
=
0
(n+1)
, and let
L
n
be the class of degrees
a
≤
0′
such that
a
n
=
0
n
. (Observe that
H
i
⊆
L
j
and
L
i
⊆
L
j
, whenever
i
≤
j
, and
H
i
∩
L
j
= ∅ for all
i
and
j.
) The result mentioned above may now be restated in the form that every minimal degree
a
≤ 0′ is in
L
2
. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree
a
< 0′ is in
H
1
. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree
a
< 0′
which is not in
L
1
. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree
a
< 0′
in
L
1
. Thus each minimal degree
a < 0′
lies in exactly one of the two classes
L
1
L
2
–
L
1
and each of the classes contains minimal degrees.
Our results are not restricted to the degrees below
0′
. We show in fact that every minimal degree
a
satisfies
a″
= (
a
∪
0′
)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below
0′
to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let
GH
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n)
, and let
GL
n
be the class of degrees
a
such that
a
(n)
= (a ∪ 0′)
(n−1)
.]]></abstract><cop>New York, USA</cop><pub>Cambridge University Press</pub><doi>10.2307/2273510</doi><tpages>10</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-4812 |
| ispartof | The Journal of symbolic logic, 1978-12, Vol.43 (4), p.715-724 |
| issn | 0022-4812 1943-5886 |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183740317 |
| source | Periodicals Index Online; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | Degree of unsolvability Eigenfunctions Mathematical functions Mathematical logic Mathematical sets Recursive functions |
| title | Double jumps of minimal degrees |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-06T14%3A01%3A39IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_proje&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Double%20jumps%20of%20minimal%20degrees&rft.jtitle=The%20Journal%20of%20symbolic%20logic&rft.au=Jockusch,%20Carl%20G.&rft.date=1978-12-01&rft.volume=43&rft.issue=4&rft.spage=715&rft.epage=724&rft.pages=715-724&rft.issn=0022-4812&rft.eissn=1943-5886&rft_id=info:doi/10.2307/2273510&rft_dat=%3Cjstor_proje%3E2273510%3C/jstor_proje%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1290590560&rft_id=info:pmid/&rft_jstor_id=2273510&rfr_iscdi=true |