User blog comment:Ikosarakt1/Extension of Chris Bird's array notation./@comment-5529393-20130307073923/@comment-5529393-20130308011139

I believe there is a problem starting from [1 [2\2] 2 \_[1,2] 2], which I believe has level ϑ(e_(Ω+1)) + (e_0)^w rather than ϑ(e_(Ω+1))^2.

More generally, I believe that when X is notation below [1 \_[1,2] 2], [X \_[1,2] 2] will have ordinal ϑ(e_(Ω+1)) + a, where a is the ordinal associated with X. So [1 \_[1,2] 1 \_[1,2] 2] will have ordinal ϑ(e_(Ω+1))^2, and [1 [2 \_[2,2] 2] 2] has level ϑ(e_(Ω+1)^w, not ϑ(e_(Ω+1) 2).  So we have to dither at low levels for a while, which make sense we are returning to low levels of nesting.

I'm guessing that [1 \_[1,3] 2] will have ordinal ϑ(e_(Ω+1) + 1), and in general [1 \_[1, 2+X] 2] will have ordinal ϑ(e_(Ω+1) + a) where a is the ordinal for X. After that things get complicated.

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<span style="vertical-align:0px;display:inline-block;position:relative;line-height:normal;white-space:nowrap;width:178px;height:0px;font-size:16px;"><span style="vertical-align:0px;position:absolute;clip:rect(26.6px16250px55.5px-7.1px);top:-45px;left:0px;"> <span class="mo" id="MathJax-Span-319" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-320" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-321" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-322" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mi" id="MathJax-Span-323" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖ <span class="mn" id="MathJax-Span-324" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-325" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mn" id="MathJax-Span-326" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mi" id="MathJax-Span-328" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖    <span class="mo" id="MathJax-Span-331" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-332" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-333" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">, <span class="mn" id="MathJax-Span-334" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-335" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">]      <span class="mn" id="MathJax-Span-336" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-337" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-338" style="padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">→ <span class="mi" id="MathJax-Span-339" style="font-style:italic;padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ϑ <span class="mo" id="MathJax-Span-340" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">(  <span style="vertical-align:0px;position:absolute;clip:rect(30.7px16250px44.4px-8.6px);top:-41px;left:0px;"><span class="mi" id="MathJax-Span-342" style="font-style:italic;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ε    <span class="mi" id="MathJax-Span-345" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">Ω <span class="mo" id="MathJax-Span-346" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">+ <span class="mn" id="MathJax-Span-347" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1       <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mo" id="MathJax-Span-349" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">)  <span class="mn" id="MathJax-Span-350" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2

<span style="vertical-align:0px;display:inline-block;position:relative;line-height:normal;white-space:nowrap;width:178px;height:0px;font-size:16px;"><span style="vertical-align:0px;position:absolute;clip:rect(26.6px16250px55.5px-7.1px);top:-45px;left:0px;">  <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mi" id="MathJax-Span-328" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖    <span class="mo" id="MathJax-Span-331" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-332" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-333" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">, <span class="mn" id="MathJax-Span-334" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-335" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">]      <span class="mn" id="MathJax-Span-336" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-337" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-338" style="padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">→ <span class="mi" id="MathJax-Span-339" style="font-style:italic;padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ϑ <span class="mo" id="MathJax-Span-340" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">(  <span style="vertical-align:0px;position:absolute;clip:rect(30.7px16250px44.4px-8.6px);top:-41px;left:0px;"><span class="mi" id="MathJax-Span-342" style="font-style:italic;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ε    <span class="mi" id="MathJax-Span-345" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">Ω <span class="mo" id="MathJax-Span-346" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">+ <span class="mn" id="MathJax-Span-347" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1       <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mo" id="MathJax-Span-349" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">)  <span class="mn" id="MathJax-Span-350" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2

<span style="border-left-style:solid;vertical-align:-8.5px;display:inline-block;position:static;line-height:normal;white-space:nowrap;overflow:hidden;width:0px;height:24.9px;">​

<span style="vertical-align:0px;display:inline-block;position:relative;line-height:normal;white-space:nowrap;width:178px;height:0px;font-size:16px;"><span style="vertical-align:0px;position:absolute;clip:rect(26.6px16250px55.5px-7.1px);top:-45px;left:0px;"> <span class="mo" id="MathJax-Span-319" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-320" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-321" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-322" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mi" id="MathJax-Span-323" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖ <span class="mn" id="MathJax-Span-324" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-325" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mn" id="MathJax-Span-326" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mi" id="MathJax-Span-328" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖    <span class="mo" id="MathJax-Span-331" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-332" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-333" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">, <span class="mn" id="MathJax-Span-334" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-335" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">]      <span class="mn" id="MathJax-Span-336" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-337" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-338" style="padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">→ <span class="mi" id="MathJax-Span-339" style="font-style:italic;padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ϑ <span class="mo" id="MathJax-Span-340" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">(  <span style="vertical-align:0px;position:absolute;clip:rect(30.7px16250px44.4px-8.6px);top:-41px;left:0px;"><span class="mi" id="MathJax-Span-342" style="font-style:italic;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ε    <span class="mi" id="MathJax-Span-345" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">Ω <span class="mo" id="MathJax-Span-346" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">+ <span class="mn" id="MathJax-Span-347" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1       <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mo" id="MathJax-Span-349" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">)  <span class="mn" id="MathJax-Span-350" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2

<span style="vertical-align:0px;display:inline-block;position:relative;line-height:normal;white-space:nowrap;width:178px;height:0px;font-size:16px;"><span style="vertical-align:0px;position:absolute;clip:rect(26.6px16250px55.5px-7.1px);top:-45px;left:0px;"> <span class="mo" id="MathJax-Span-319" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-320" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-321" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-322" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mi" id="MathJax-Span-323" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖ <span class="mn" id="MathJax-Span-324" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-325" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mn" id="MathJax-Span-326" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mi" id="MathJax-Span-328" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">∖    <span class="mo" id="MathJax-Span-331" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">[ <span class="mn" id="MathJax-Span-332" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1 <span class="mo" id="MathJax-Span-333" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">, <span class="mn" id="MathJax-Span-334" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-335" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">]      <span class="mn" id="MathJax-Span-336" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">2 <span class="mo" id="MathJax-Span-337" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-338" style="padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">→ <span class="mi" id="MathJax-Span-339" style="font-style:italic;padding-left:4.5px;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ϑ <span class="mo" id="MathJax-Span-340" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">(  <span style="vertical-align:0px;position:absolute;clip:rect(30.7px16250px44.4px-8.6px);top:-41px;left:0px;"><span class="mi" id="MathJax-Span-342" style="font-style:italic;vertical-align:0px;display:inline;position:static;font-family:MathJax_Math;">ε    <span class="mi" id="MathJax-Span-345" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">Ω <span class="mo" id="MathJax-Span-346" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">+ <span class="mn" id="MathJax-Span-347" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">1       <span style="vertical-align:0px;position:absolute;clip:rect(29.8px16250px52.1px-8.1px);top:-45px;left:0px;"><span class="mo" id="MathJax-Span-349" style="vertical-align:0px;display:inline;position:static;font-family:MathJax_Main;">)  <span class="mn" id="MathJax-Span-350" style="vertical-align:0px;display:inline;position:static;font-size:11px;font-family:MathJax_Main;">2